Variedades não matriciais em certas classes de álgebras não associativas

(1)

❱❛r✐❡❞❛❞❡s ♥ã♦ ♠❛tr✐❝✐❛✐s ❡♠ ❝❡rt❛s

❝❧❛ss❡s ❞❡ á❧❣❡❜r❛s ♥ã♦ ❛ss♦❝✐❛t✐✈❛s

❱✐♥✐❝✐✉s ❙♦✉③❛ ❇✐tt❡♥❝♦✉rt

❚❡s❡ ❛♣r❡s❡♥t❛❞❛

❛♦

■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛

❞❛

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦

♣❛r❛

♦❜t❡♥çã♦ ❞♦ tít✉❧♦

❞❡

❉♦✉t♦r ❡♠ ❈✐ê♥❝✐❛s

Pr♦❣r❛♠❛✿ ▼❛t❡♠át✐❝❛

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ■✈❛♥ P✳ ❙❤❡st❛❦♦✈

❉✉r❛♥t❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦ ♦ ❛✉t♦r r❡❝❡❜❡✉ ❛✉①í❧✐♦ ✜♥❛♥❝❡✐r♦ ❞❛ ❈◆Pq

(2)

❱❛r✐❡❞❛❞❡s ♥ã♦ ♠❛tr✐❝✐❛✐s

❡♠ ❝❡rt❛s ❝❧❛ss❡s ❞❡ á❧❣❡❜r❛s ♥ã♦ ❛ss♦❝✐❛t✐✈❛s

❊st❛ ✈❡rsã♦ ❞❛ t❡s❡ ❝♦♥té♠ ❛s ❝♦rr❡çõ❡s ❡ ❛❧t❡r❛çõ❡s s✉❣❡r✐❞❛s ♣❡❧❛ ❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛ ❞✉r❛♥t❡ ❛ ❞❡❢❡s❛ ❞❛ ✈❡rsã♦ ♦r✐❣✐♥❛❧ ❞♦ tr❛❜❛❧❤♦✱ r❡❛❧✐③❛❞❛ ❡♠ ✵✸✴✵✺✴✷✵✶✻✳ ❯♠❛ ❝ó♣✐❛ ❞❛ ✈❡rsã♦ ♦r✐❣✐♥❛❧ ❡stá ❞✐s♣♦♥í✈❡❧ ♥♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✳

❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛✿

• Pr♦❢✳ ❉r✳ ■✈❛♥ P❛✈❧♦✈✐❝❤ ❙❤❡st❛❦♦✈ ✭♦r✐❡♥t❛❞♦r✮ ✲ ■▼❊✲❯❙P • Pr♦❢❛✳ ❉r❛✳ ▲✉❝✐❛ ❙❛t✐❡ ■❦❡♠♦t♦ ▼✉r❛❦❛♠✐ ✲ ■▼❊✲❯❙P • Pr♦❢✳ ❉r✳ ❆❧❡①❛♥❞r ❑♦r♥❡✈ ✲ ❯❋❆❇❈

• Pr♦❢✳ ❉r✳ ❱✐❝t♦r P❡tr♦❣r❛❞s❦✐② ✲ ❯♥❇

(3)

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ Pr♦❢✳ ■✈❛♥ ❙❤❡st❛❦♦✈✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ❛t❡♥çã♦ ❡ ❞❡❞✐❝❛çã♦ ❡♠♣r❡❡♥❞✐❞❛s ♥❡st❡ tr❛❜❛❧❤♦✱ ❛❧é♠ ❞❡ ❝♦♥tr✐❜✉✐r ❣r❛♥❞❡♠❡♥t❡ ♣❛r❛ ♦ ❛♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ♠❛t❡♠át✐❝❛✳

❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ▲♦✉r✐✈❛❧✱ ▲í❝✐❛ ❡ ❱✐✈✐❛♥❡✱ ♣❡❧♦ s✉♣♦rt❡✱ ❛♣♦✐♦ ❡ ❝❛r✐♥❤♦✳ ❆❣r❛❞❡ç♦ à ♠✐♥❤❛ q✉❡r✐❞❛ ❆♥❣❡❧❛✱ ✉♠ ❜r❛ç♦ ❢♦rt❡ ♥❡st❛ r❡t❛ ✜♥❛❧ ❞❛ t❡s❡✳

❆❣r❛❡ç♦ ❛♦s ♠❡✉s ♣❛r❡♥t❡s✱ tã♦ tã♦ ❞✐st❛♥t❡s ❣❡♦❣r❛✜❝❛♠❡♥t❡✱ ♣❡❧❛ t♦r❝✐❞❛ ❡ ♣❡❧❛ ❢♦rç❛✦

❆❣r❛❞❡ç♦ ❛♦ ❝♦r♣♦ ❞♦❝❡♥t❡ ❞♦ ■▼❊ ♣❡❧❛ ❞❡❞✐❝❛çã♦ ♥❛s ❞✐s❝✐♣❧✐♥❛s✱ ♣❛❧❡str❛s ❡ ❝✉rs♦s ♠✐♥✐str❛❞♦s✳

❆❣r❛❞❡ç♦ ❛♦ ♣r♦❢❡ss♦r ❚❤✐❡rr② ▲♦❜ã♦✱ ♣♦r ❤❛✈❡r ♠❡ ❛♣r❡s❡♥t❛❞♦ ♦s ✏♣r✐♠❡✐r♦s ♣❛ss♦s✑ ❞❛ ♣❡sq✉✐s❛ ♠❛t❡♠át✐❝❛✳

❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❛♠✐❣♦s ❡♠ ❙ã♦ P❛✉❧♦✱ ❡♠ ❡s♣❡❝✐❛❧ ■r❡♠❛r✱ ❊s❞r❛s✱ ▲✉❝❛s✱ ❆❧❡①✱ ❊❞✉❛r❞♦ ❡ ❆❧❡ss❛♥❞r❛✱ ♣❡❧❛ ❢♦rç❛ ❡ ♣❡❧❛s ❢❡st❛s✦

❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❛♠✐❣♦s t♦r❝❡❞♦r❡s ❞♦ ❱✐tór✐❛ ✭♠✐♥❤❛ ✈✐❞❛✦✮ r❡s✐❞❡♥t❡s ❡♠ ❙ã♦ P❛✉❧♦✱ ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ ❛❧❡❣r✐❛✳

❆❣r❛❞❡ç♦ às ❢❛♠í❧✐❛s ❖❧✐✈❡✐r❛ ❡ ▼❡❧♦✱ ♣♦r ♣❛rt✐❧❤❛r❡♠ ✉♠ ♣♦✉❝♦ ❞❡ss❡ ♣r♦❝❡ss♦ ❝♦♠✐❣♦✳ ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s q✉❡r✐❞♦s ❡①✲❛❧✉♥♦s ❡ ❝♦❧❡❣❛s ❞❛ ❊❇❊■✳

❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ❛♠✐❣♦ ▲✉✐③ ▼❛r❝✐♦✱ ♣♦r ❛♣r❡s❡♥t❛r✲♠❡ ♦ ❧❛❞♦ ♥❡❣r♦ ❞❛ ❋♦rç❛✳ ❆❣r❛❞❡ç♦ ❛♦s ❛♠✐❣♦s ❞❛ ♠✐♥❤❛ t❡rr❛ ♥❛t❛❧✱ ♣♦r t♦r❝❡r❡♠ t❛♥t♦✱ ❛✐♥❞❛ q✉❡ à ❞✐stâ♥❝✐❛✳

(4)

(5)

❘❡s✉♠♦

❇■❚❚❊◆❈❖❯❘❚✱ ❱✳ ❙✳ ❱❛r✐❡❞❛❞❡s ♥ã♦ ♠❛tr✐❝✐❛✐s ❡♠ ❝❡rt❛s ❝❧❛ss❡s ❞❡ á❧❣❡❜r❛s ♥ã♦ ❛ss♦❝✐❛t✐✈❛s✳ ✷✵✶✻✳ ✽✶ ❢✳ ❚❡s❡ ✭❉♦✉t♦r❛❞♦✮ ✲ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛✱ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ❙ã♦ P❛✉❧♦✱ ✷✵✶✻✳

❯♠❛ ✈❛r✐❡❞❛❞❡ M ❞❡ á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s é ❞✐t❛ s❡r ✏♥ã♦ ♠❛tr✐❝✐❛❧✑ s❡ F2 ∈ M✱ ❡♠/ q✉❡F2 é ♦ ❛♥❡❧ ❞❛s ♠❛tr✐③❡s q✉❛❞r❛❞❛s ❞❡ ♦r❞❡♠ ✷ s♦❜r❡F✳ ❱✳ ▲❛t②s❤❡✈ ✐♥tr♦❞✉③✐✉ ❡st❛s ✈❛r✐❡❞❛❞❡s ❡♠ ❬▲❛t✼✼❪✳ ❆ r❡s♣❡✐t♦ ❞❡st❛ ❞❡✜♥✐çã♦✱ ♦✉tr❛s ❝❛r❛❝t❡r✐③❛çõ❡s ❡q✉✐✈❛❧❡♥t❡s ♣❛r❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ ♥ã♦ ♠❛tr✐❝✐❛❧ ❢♦r❛♠ ♦❜t✐❞❛s✱ ♣♦r ❡①❡♠♣❧♦✱ ❛♦ ❝♦♥s✐❞❡r❛r ❡❧❡♠❡♥t♦s ❛❧❣é❜r✐❝♦s ❬❈❡❦✼✾❪ ❡ ♥✐❧♣♦t❡♥t❡s ❬▼P❘✷✵✶✶❪✳ ❱❛r✐❡❞❛❞❡s ♥ã♦ ♠❛tr✐❝✐❛✐s sã♦ ❡st✉❞❛❞❛s ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦ ❝❛s♦ s♦❜r❡ ♦s ❝♦r♣♦s ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ③❡r♦ ♣❛r❛ á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s✳

❆ t❡♦r✐❛ ❣❡r❛❧ ❞❡ ✈❛r✐❡❞❛❞❡s ❞❡ á❧❣❡❜r❛s✱ ❡♥tr❡t❛♥t♦✱ ♥ã♦ ❡stá r❡str✐t❛ à ❝❧❛ss❡ ❞❛s á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s✳ ❆❧é♠ ❞❛s á❧❣❡❜r❛s ❞❡ ▲✐❡✱ ❡♥tr❡ ❛s ♠✉✐t❛s ❝❧❛ss❡s ❞❡ á❧❣❡❜r❛s ♥ã♦ ❛ss♦❝✐❛t✐✈❛s✱ ♥ós ❞❡st❛❝❛♠♦s ❛s á❧❣❡❜r❛s ❛❧t❡r♥❛t✐✈❛s✱ ❛s ❞❡ ❏♦r❞❛♥ ❡ ❛s ❞❡ ❏♦r❞❛♥ ♥ã♦ ❝♦♠✉t❛t✐✈❛s✳ ❊st❛s ❝❧❛ss❡s ❞❡ á❧❣❡❜r❛s tê♠ ♠✉✐t❛s ❝♦♥❡①õ❡s ❡ ❛♣❧✐❝❛çõ❡s ❛ ❞✐✈❡rs❛s ár❡❛s ❞❛ ▼❛t❡♠át✐❝❛ ❡ ❞❛ ❋ís✐❝❛ ❡ tê♠ ✉♠❛ t❡♦r✐❛ ❡str✉t✉r❛❧ ❜❡♠ ❞❡s❡♥✈♦❧✈✐❞❛✱ ❛ss✐♠ ❝♦♠♦ ❛ ❝❧❛ss❡ ❞❛s á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s✳

❖ ❝♦♥❝❡✐t♦ ❞❡ ✏✈❛r✐❡❞❛❞❡ ♥ã♦ ♠❛tr✐❝✐❛❧✑ ♣♦❞❡ s❡r r❡❢♦r♠✉❧❛❞♦ ♣❛r❛ ❛s ❝❧❛ss❡s ❞❡ á❧✲ ❣❡❜r❛s s✉♣r❛❝✐t❛❞❛s ❡ ♥♦ss♦ tr❛❜❛❧❤♦ ❝♦♥s✐st❡ ❡♠ ❛❞❛♣t❛r✱ ❡st❡♥❞❡r ♦✉ ❣❡♥❡r❛❧✐③❛r ❛❧❣✉♥s r❡s✉❧t❛❞♦s✱ ❝♦♥❢♦r♠❡ ♠❡♥❝✐♦♥❛❞♦✱ ♣❛r❛ ✈❛r✐❡❞❛❞❡s ♥ã♦ ♠❛tr✐❝✐❛✐s ♥❡st❛s ❝❧❛ss❡s ❞❡ á❧❣❡✲ ❜r❛s✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ á❧❣❡❜r❛✱ ♥ã♦ ♠❛tr✐❝✐❛❧✱ ✈❛r✐❡❞❛❞❡✱ P■✱ ♥ã♦ ❛ss♦❝✐❛t✐✈♦✳

(6)

(7)

❆❜str❛❝t

❇■❚❚❊◆❈❖❯❘❚✱ ❱✳ ❙✳ ◆♦♥♠❛tr✐① ✈❛r✐❡t✐❡s ✐♥ ❝❡rt❛✐♥ ❝❧❛ss❡s ♦❢ ♥♦♥ ❛ss♦❝✐❛✲ t✐✈❡ ❛❧❣❡❜r❛s✳ ✷✵✶✻✳ ✽✶ s✳ ❚❡s❡ ✭❉♦✉t♦r❛❞♦✮ ✲ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛✱ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ❙ã♦ P❛✉❧♦✱ ✷✵✶✻✳

❆ ✈❛r✐❡t② M♦❢ ❛ss♦❝✐❛t✐✈❡ ❛❧❣❡❜r❛s ✭♦✈❡r ❛ ✜❡❧❞ F✮ ✐s ❝❛❧❧❡❞ ✏♥♦♥♠❛tr✐①✑ ✐❢ F2 _∈/ M✱ ✇❤❡r❡ F2 ✐s t❤❡ ✉s✉❛❧ ♠❛tr✐① ❛❧❣❡❜r❛ ♦❢ s❡❝♦♥❞ ♦r❞❡r ♦✈❡r F✳ ❱✳ ▲❛t②s❤❡✈ ✐♥tr♦❞✉❝❡❞ t❤❡s❡ ✈❛r✐❡t✐❡s ✐♥ ❬▲❛t✼✼❪✳ ❈♦♥❝❡r♥✐♥❣ t❤✐s ❞❡✜♥✐t✐♦♥✱ ♦t❤❡r ❡q✉✐✈❛❧❡♥t ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ❢♦r ❛ ♥♦♥♠❛tr✐① ✈❛r✐❡t② ✇❡r❡ ♦❜t❛✐♥❡❞✱ ❢♦r ✐♥st❛♥❝❡✱ ❜② ❝♦♥s✐❞❡r✐♥❣ ❛❧❣❡❜r❛✐❝ ❬❈❡❦✼✾❪ ❛♥❞ ♥✐❧♣♦t❡♥t ❬▼P❘✷✵✶✶❪ ❡❧❡♠❡♥ts✳ ◆♦♥✲♠❛tr✐① ✈❛r✐❡t✐❡s ❛r❡ st✉❞✐❡❞ ♠❛✐♥❧② ✐♥ t❤❡ ❝❛s❡ ♦❢ ❝❤❛r❛❝t❡r✐st✐❝ ③❡r♦ ❢♦r ❛ss♦❝✐❛t✐✈❡ ❛❧❣❡❜r❛s✳

❍♦✇❡✈❡r✱ t❤❡ ❣❡♥❡r❛❧ t❤❡♦r② ♦❢ ✈❛r✐❡t✐❡s ♦❢ ❛❧❣❡❜r❛s ✐s ♥♦t r❡str✐❝t❡❞ t♦ t❤❡ ❝❧❛ss ♦❢ ❛s✲ s♦❝✐❛t✐✈❡ ❛❧❣❡❜r❛s✳ ■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ▲✐❡ ❛❧❣❡❜r❛s✱ ❛♠♦♥❣ ♠❛♥② ❝❧❛ss❡s ♦❢ ♥♦♥ ❛ss♦❝✐❛t✐✈❡ ❛❧❣❡❜r❛s✱ ✇❡ ❤✐❣❤❧✐❣❤t t❤❡ ❛❧t❡r♥❛t✐✈❡✱ t❤❡ ❏♦r❞❛♥ ❛♥❞ t❤❡ ♥♦♥ ❝♦♠♠✉t❛t✐✈❡ ❏♦r❞❛♥ ❛❧❣❡✲ ❜r❛s✳ ❚❤❡s❡ ❝❧❛ss❡s ♦❢ ❛❧❣❡❜r❛s ❤❛✈❡ ♠❛♥② ❝♦♥♥❡①✐♦♥s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s t♦ s❡✈❡r❛❧ ❛r❡❛s ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ P❤②s✐❝s ❛♥❞ ❤❛✈❡ ❛ ✇❡❧❧✲❞❡✈❡❧♦♣❡❞ str✉❝t✉r❛❧ t❤❡♦r②✱ ❛s ✐♥ t❤❡ ❝❧❛ss ♦❢ ❛ss♦❝✐❛t✐✈❡ ❛❧❣❡❜r❛s✳

❚❤❡ ❝♦♥❝❡♣t ♦❢ ✏♥♦♥♠❛tr✐① ✈❛r✐❡t②✑ ❝❛♥ ❜❡ r❡❢♦r♠✉❧❛t❡❞ ✐♥ t❤❡ ❝❧❛ss❡s ♦❢ ❛❧❣❡❜r❛s ❛❜♦✈❡ ❛♥❞ ♦✉r ✇♦r❦ ✐s t♦ ❛❞❛♣t✱ ❡①t❡♥❞ ♦r ❣❡♥❡r❛❧✐③❡ s♦♠❡ r❡s✉❧ts✱ ❛s ♠❡♥t✐♦♥❡❞✱ ❢♦r ♥♦♥✲♠❛tr✐① ✈❛r✐❡t✐❡s ✐♥ t❤❡s❡ ❝❧❛ss❡s ♦❢ ❛❧❣❡❜r❛s✳

❑❡②✇♦r❞s✿ ❛❧❣❡❜r❛✱ ♥♦♥♠❛tr✐①✱ ✈❛r✐❡t②✱ P■✱ ♥♦♥ ❛ss♦❝✐❛t✐✈❡✳

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❙✉♠ár✐♦

▲✐st❛ ❞❡ ❙í♠❜♦❧♦s ✐①

✶ ■♥tr♦❞✉çã♦ ✶

✶✳✶ ❈♦♥s✐❞❡r❛çõ❡s Pr❡❧✐♠✐♥❛r❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶

✶✳✷ ❖❜❥❡t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶

✶✳✸ ❈♦♥tr✐❜✉✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷

✶✳✹ ❖r❣❛♥✐③❛çã♦ ❞♦ ❚r❛❜❛❧❤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷

✷ ❊❧❡♠❡♥t♦s ❣❡r❛✐s ❡♠ á❧❣❡❜r❛s ✸

✷✳✶ ❉❡✜♥✐çõ❡s ❜ás✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸

✷✳✷ ❱❛r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾

✷✳✸ ❊❧❡♠❡♥t♦s ❞❛ ❚❡♦r✐❛ ●❡r❛❧ ❞❡ ❘❛❞✐❝❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✷✳✸✳✶ ❊①❡♠♣❧♦s ❞❡ r❛❞✐❝❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺

✷✳✹ P■✲á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✷✳✺ ❱❛r✐❡❞❛❞❡s ♥ã♦ ♠❛tr✐❝✐❛✐s ❡♠Assoc ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✸ ➪❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥ ✷✼

✸✳✶ ❉❡✜♥✐çõ❡s ❡ ❡①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

✸✳✶✳✶ P■✲á❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✸✳✷ ❘❛❞✐❝❛✐s ❡♠ Jord ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✸✳✸ ❱❛r✐❡❞❛❞❡s ♥ã♦ ♠❛tr✐❝✐❛✐s ❡♠Jord ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻

✹ ➪❧❣❡❜r❛s ❛❧t❡r♥❛t✐✈❛s ✹✺

✹✳✶ ❉❡✜♥✐çõ❡s ❡ ❡①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

✹✳✷ ❘❛❞✐❝❛✐s ❡♠ Alt✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

✹✳✸ ❱❛r✐❡❞❛❞❡s ♥ã♦ ♠❛tr✐❝✐❛✐s ❡♠Alt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽

✹✳✹ ❖ ❝❛s♦ ♥ã♦ ♠❛tr✐❝✐❛❧ ♣❛r❛ ✭✲✶✱✶✮✲á❧❣❡❜r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

✺ ➪❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥ ♥ã♦✲❝♦♠✉t❛t✐✈❛s ✺✼

✺✳✶ ❊①❡♠♣❧♦s ❞❡ á❧❣❡❜r❛s ❡♠ NCJ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

✺✳✶✳✶ ➪❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥ ♥ã♦ ❝♦♠✉t❛t✐✈❛s ❡str✐t❛♠❡♥t❡ ♣r✐♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

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✈✐✐✐ ❙❯▼➪❘■❖

✺✳✷ ➪❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥ ♥ã♦ ❝♦♠✉t❛t✐✈❛s ❛❞♠✐ssí✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷

✺✳✸ ❱❛r✐❡❞❛❞❡s ♥ã♦ ♠❛tr✐❝✐❛✐s ❡♠ NCJ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼

✻ ❈♦♥❝❧✉sõ❡s ✼✺

✻✳✶ ❙✉❣❡stõ❡s ♣❛r❛ P❡sq✉✐s❛s ❋✉t✉r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺

✻✳✷ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼

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▲✐st❛ ❞❡ ❙í♠❜♦❧♦s

(x, y, z) ❛ss♦❝✐❛❞♦r✿ (xy)z₋x(yz)

[x, y] ❝♦♠✉t❛❞♦r✿ xy₋yx

x_◦y ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❏♦r❞❛♥✿ 1₂(xy+yx)

A(+) (A,+,_◦)

(x, y, z)+ ❛ss♦❝✐❛❞♦r ❡♠ A(+)

V✱ M ✈❛r✐❡❞❛❞❡s ❞❡ á❧❣❡❜r❛s

Assoc ✈❛r✐❡❞❛❞❡ ❞❛s á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s Com ✈❛r✐❡❞❛❞❡ ❞❛s á❧❣❡❜r❛s ❝♦♠✉t❛t✐✈❛s Jord ✈❛r✐❡❞❛❞❡ ❞❛s á❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥ Alt ✈❛r✐❡❞❛❞❡ ❞❛s á❧❣❡❜r❛s ❛❧t❡r♥❛t✐✈❛s

NCJ ✈❛r✐❡❞❛❞❡ ❞❛s á❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥ ♥ã♦ ❝♦♠✉t❛t✐✈❛s

F ❝♦r♣♦ ✭❣❡r❛❧♠❡♥t❡ ✐♥✜♥✐t♦✮

Φ ❛♥❡❧ ❞❡ ❡s❝❛❧❛r❡s ♣❛r❛ á❧❣❡❜r❛s

N ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s

Z ❛♥❡❧ ❞♦s ✐♥t❡✐r♦s

Q ❝♦r♣♦ ❞♦s r❛❝✐♦♥❛✐s

R ❝♦r♣♦ ❞♦s r❡❛✐s

C ❝♦r♣♦ ❞♦s ❝♦♠♣❧❡①♦s ✭✉s✉❛✐s✮

H ❛♥❡❧ ❞❡ ❞✐✈✐sã♦ ❞♦s q✉❛tér♥✐♦s ✭✉s✉❛✐s✮

O á❧❣❡❜r❛ ❞♦s ♦❝tô♥✐♦s ✭✉s✉❛✐s✮

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❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❯♠❛ ✈❛r✐❡❞❛❞❡ M ❞❡ á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s é ❞✐t❛ s❡r ✏♥ã♦ ♠❛tr✐❝✐❛❧✑ s❡ F2 ∈ M✱/ ❡♠ q✉❡ F2 é ♦ ❛♥❡❧ ❞❛s ♠❛tr✐③❡s q✉❛❞r❛❞❛s ❞❡ ♦r❞❡♠ ✷ s♦❜r❡ F✳ ❚❛✐s ✈❛r✐❡❞❛❞❡s ❢♦r❛♠ ✐♥tr♦❞✉③✐❞❛s ♣♦r ▲❛t②s❤❡✈ ❡♠ ✶✾✼✼✱ ❬▲❛t✼✼❪✳

❆ t❡♦r✐❛ ❞❡ ✈❛r✐❡❞❛❞❡s ❞❡ á❧❣❡❜r❛s ♥ã♦ ❡stá r❡str✐t❛ à ❝❧❛ss❡ ❞❛s á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✲ ✈❛s✳ ❉❡♥tr❡ ❛s ♠✉✐t❛s ❝❧❛ss❡s ❞❡ á❧❣❡❜r❛s ♥ã♦ ❛ss♦❝✐❛t✐✈❛s✱ ♥ós ❞❡st❛❝❛♠♦s ❛s á❧❣❡❜r❛s ❛❧t❡r♥❛t✐✈❛s✱ ❛s á❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥ ❡ ❛s á❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥ ♥ã♦ ❝♦♠✉t❛t✐✈❛s✳ ❊st❛s ❝❧❛s✲ s❡s ❞❡ á❧❣❡❜r❛s tê♠ ♠✉✐t❛s ❝♦♥❡①õ❡s ❡ ❛♣❧✐❝❛çõ❡s ❛ ❞✐✈❡rs❛s ár❡❛s ❞❛ ▼❛t❡♠át✐❝❛ ❡ ❞❛ ❋ís✐❝❛ ❡ tê♠ ✉♠❛ t❡♦r✐❛ ❡str✉t✉r❛❧ ❜❡♠ ❞❡s❡♥✈♦❧✈✐❞❛✱ ❛ss✐♠ ❝♦♠♦ ❛ ❝❧❛ss❡ ❞❛s á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s✳

❖ ❝♦♥❝❡✐t♦ ❞❡ ✏✈❛r✐❡❞❛❞❡ ♥ã♦ ♠❛tr✐❝✐❛❧✑ ♣♦❞❡ s❡r r❡❢♦r♠✉❧❛❞♦ ♣❛r❛ ❛s ❝❧❛ss❡s ❞❡ á❧✲ ❣❡❜r❛s s✉♣r❛❝✐t❛❞❛s ❡ ♥♦ss♦ tr❛❜❛❧❤♦ ❝♦♥s✐st❡ ❡♠ ❛❞❛♣t❛r✱ ❡st❡♥❞❡r ♦✉ ❣❡♥❡r❛❧✐③❛r ❛❧❣✉♥s r❡s✉❧t❛❞♦s✱ ❝♦♥❢♦r♠❡ ♠❡♥❝✐♦♥❛❞♦✱ ♣❛r❛ ✈❛r✐❡❞❛❞❡s ♥ã♦ ♠❛tr✐❝✐❛✐s ♥❡st❛s ❝❧❛ss❡s ❞❡ á❧❣❡✲ ❜r❛s✳

✶✳✶ ❈♦♥s✐❞❡r❛çõ❡s Pr❡❧✐♠✐♥❛r❡s

❆♦ ❧♦♥❣♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ✉♠❛ á❧❣❡❜r❛ s❡rá ✉♠❛ Φ✲á❧❣❡❜r❛✱ ❡♠ q✉❡ Φ é ✉♠ ❛♥❡❧ ❞❡ ❡s❝❛❧❛r❡s✱ ✐st♦ é✱ Φ é ✉♠ ❛♥❡❧ ❛ss♦❝✐❛t✐✈♦✱ ❝♦♠✉t❛t✐✈♦ ❡ ❝♦♠ ❡❧❡♠❡♥t♦ ✐❞❡♥t✐❞❛❞❡✳

❆ ❧❡tr❛F s❡rá ✉s❛❞❛ ♣❛r❛ ❞❡♥♦t❛r ❝♦r♣♦s ✭q✉❛s❡ s❡♠♣r❡✱ ❝♦r♣♦s ✐♥✜♥✐t♦s✮✳ ◆♦t❛r❡♠♦s ❛s ♠❛tr✐③❡s 2_×2s♦❜r❡ F ♣♦r F2 ❡ ❛s ♠❛tr✐③❡s ❤❡r♠✐t✐❛♥❛s s♦❜r❡ F2 ♣♦r_H2✳

✶✳✷ ❖❜❥❡t✐✈♦s

◆♦ss♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ é ♦❢❡r❡❝❡r ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ♥ã♦ ♠❛tr✐❝✐❛❧ ♣❛r❛ ✈❛r✐❡❞❛❞❡s ❡♠ ❛❧❣✉♠❛s ❝❧❛ss❡s ❞❡ á❧❣❡❜r❛s ♥ã♦ ❛ss♦❝✐❛t✐✈❛s✳ ❈♦♠♦ ❤á ♠✉✐t❛s ❝❧❛ss❡s ❞❡ á❧❣❡❜r❛s ♥ã♦ ❛ss♦❝✐❛t✐✈❛s ❝♦♥❤❡❝✐❞❛s ❡ ❡st✉❞❛❞❛s ♥❛ ❧✐t❡r❛t✉r❛✱ ♥♦s ❛t❡r❡♠♦s ❛ q✉❛tr♦ ❝❧❛ss❡s ❞❡st❡ t✐♣♦✿ ❛ ❝❧❛ss❡ ❞❛s á❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥✱ ❛ ❝❧❛ss❡ ❞❛s á❧❣❡❜r❛s ❛❧t❡r♥❛t✐✈❛s✱ ❛ ❝❧❛ss❡ ❞❛s á❧❣❡❜r❛s ✭✲✶✱✶✮ ❡ ❛ ❝❧❛ss❡ ❞❛s á❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥ ♥ã♦ ❝♦♠✉t❛t✐✈❛s✳

P❛r❛ t❛♥t♦✱ ❞❡s❡♥✈♦❧✈❡r❡♠♦s ♥♦✈❛s ❝♦♥str✉çõ❡s ❡ ♥♦✈❛s té❝♥✐❝❛s ❝♦♥❝❡r♥❡♥t❡s ♣❛r❛ á❧❣❡❜r❛s ♥ã♦ ❛ss♦❝✐❛t✐✈❛s✳

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✷ ■◆❚❘❖❉❯➬➹❖ ✶✳✹

✶✳✸ ❈♦♥tr✐❜✉✐çõ❡s

❆s ♣r✐♥❝✐♣❛✐s ❝♦♥tr✐❜✉✐çõ❡s ❞❡st❡ tr❛❜❛❧❤♦ sã♦ ❛s s❡❣✉✐♥t❡s✿

• ❯♠❛ ♥♦✈❛ ♠❛♥❡✐r❛ ❞❡ ❝♦♥❝❡✐t✉❛r ✈❛r✐❡❞❛❞❡s ♥ã♦ ♠❛tr✐❝✐❛✐s❀

• ❆ ❣❡♥❡r❛❧✐③❛çã♦ ❞❡ té❝♥✐❝❛s ❡♥✈♦❧✈❡♥❞♦ ❝♦♥❝❡✐t♦s ❞❡ ✏♥ã♦ ♠❛tr✐❝✐❛❧✐❞❛❞❡✑ ♣❛r❛ á❧✲ ❣❡❜r❛s ♥ã♦ ❛ss♦❝✐❛t✐✈❛s✳

❏✉❧❣❛♠♦s t❛✐s ❝♦♥tr✐❜✉✐çõ❡s ❝♦♠♦ r❡❧❡✈❛♥t❡s ♣❛r❛ ❡st❡ tr❛❜❛❧❤♦✱ ❜❡♠ ❝♦♠♦ ♣❛r❛ ❛ ❧✐t❡r❛t✉r❛ ♠❛t❡♠át✐❝❛✳

✶✳✹ ❖r❣❛♥✐③❛çã♦ ❞♦ ❚r❛❜❛❧❤♦

◆♦ ❈❛♣ít✉❧♦ ✷✱ ❛♣r❡s❡♥t❛♠♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s r❡❧❛t✐✈♦s ❛ ❡st❡ tr❛❜❛❧❤♦✳ ■♥❞✐❝❛♠♦s ✉♠❛ ❇✐❜❧✐♦❣r❛✜❛ ❇ás✐❝❛ ♣❛r❛ ✉♠ ❧❡✐t♦r ♥ã♦ ♠✉✐t♦ ❢❛♠✐❧✐❛r✐③❛❞♦ ❝♦♠ ♦ ❛ss✉♥t♦✱ ♠❛s t❛♠✲ ❜é♠ ♣❛r❛ ✉♠ ❧❡✐t♦r ♠❛✐s ❡①♣❡r✐❡♥t❡ ❡ q✉❡ ♥❡❝❡ss✐t❛ t❡r ✉♠❛ r❡❢❡rê♥❝✐❛✳ ❆♣r❡s❡♥t❛♠♦s ♦s ♦❜❥❡t♦s ♠❛t❡♠át✐❝♦s q✉❡ s❡r✈✐rã♦ ❞❡ ❛♣♦✐♦ ❛ ❡st❛ t❡s❡✳

◆♦ ❈❛♣ít✉❧♦ ✸✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ✈❛r✐❡❞❛❞❡ ♥ã♦ ♠❛tr✐❝✐❛❧ ❡♠ á❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥✳ ❉❡ ♣♦ss❡ ❞❡st❛ ❞❡✜♥✐çã♦✱ ♣r♦✈❛♠♦s q✉❡✱ ♣❛r❛ á❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥✱ ✈❛❧❡ ♦ ❛♥á❧♦❣♦ ❞❡ ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ♥ã♦ ♠❛tr✐❝✐❛❧ ❢❡✐t❛ ♣❛r❛ ❛s ❝❧❛ss❡s ❞❛s á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s✳ ❆✐♥❞❛✱ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ♥♦✈❛ ♠❛♥❡✐r❛ ❞❡ ❝♦♥❝❡✐t✉❛r ✈❛r✐❡❞❛❞❡s ♥ã♦ ♠❛tr✐❝✐❛✐s✱ ♣♦✐s t❡♥t❛r❡♠♦s ✉♥✐✈❡rs❛❧✐③❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ ✏♥ã♦ ♠❛tr✐❝✐❛❧✑ ♣❛r❛ q✉❛❧q✉❡r ❝❧❛ss❡ ❞❡ á❧❣❡❜r❛s✳

◆♦ ❈❛♣ít✉❧♦ ✹✱ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❡①t❡♥sã♦ ✭❝♦♠ r❡s♣❡✐t♦ às á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s✮ ❞❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ✈❛r✐❡❞❛❞❡ ♥ã♦ ♠❛tr✐❝✐❛❧ ♣❛r❛ ❛s á❧❣❡❜r❛s ❛❧t❡r♥❛t✐✈❛s✳ ❆❧é♠ ❞✐ss♦✱ ❡①✐❜✐♠♦s ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ♥ã♦ ♠❛tr✐❝✐❛❧ ♣❛r❛ ✉♠❛ ❝❧❛ss❡ ❞❡ á❧❣❡❜r❛s ❛❧t❡r♥❛t✐✈❛s à ❞✐r❡✐t❛✱ ❛s á❧❣❡❜r❛s ✭✲✶✱✶✮✳

◆♦ ❈❛♣ít✉❧♦ ✺✱ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ sér✐❡ ❞❡ ♥♦✈♦s ❝♦♥❝❡✐t♦s r❡❧❛t✐✈♦s às á❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥ ♥ã♦ ❝♦♠✉t❛t✐✈❛s✱ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❢♦r♥❡❝❡r ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ♥ã♦ ♠❛tr✐❝✐❛❧✳ ❉❡ ❢❛t♦✱ ❢♦♠♦s ❝❛♣❛③❡s ❞❡ ❝❛r❛❝t❡r✐③❛r ✈❛r✐❡❞❛❞❡s ♥ã♦ ♠❛tr✐❝✐❛✐s ❡♠ ✉♠❛ ❝❧❛ss❡ ♣❛rt✐❝✉❧❛r✱ ❛ ❝❧❛ss❡ ❞❛s á❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥ ♥ã♦ ❝♦♠✉t❛t✐✈❛s ❛❞♠✐ssí✈❡✐s✱ q✉❡ é ✉♠❛ ❝❧❛ss❡ q✉❡ ❝♦♥té♠ ❛s á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s ❡ ❛s á❧❣❡❜r❛s ❝♦♠✉t❛t✐✈❛s✳

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❈❛♣ít✉❧♦ ✷

❊❧❡♠❡♥t♦s ❣❡r❛✐s ❡♠ á❧❣❡❜r❛s

◆❡st❡ ❝❛♣ít✉❧♦✱ s❡rã♦ ❧✐st❛❞♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❜ás✐❝♦s ❞❡ á❧❣❡❜r❛s✳ P❛r❛ ✉♠ ❧❡✐t♦r ♥ã♦ ❢❛♠✐❧✐❛r✐③❛❞♦ ❝♦♠ ♦ ❛ss✉♥t♦✱ r❡❝♦♠❡♥❞❛♠♦s ❛ ❧❡✐t✉r❛ ❞❡ ❬❍✉♥❣✼✹❪✱ ❬▲❛♥❣✵✷❪✱ ❬●✻✾❪ ❡ ❬❩❙❙❙✽✷❪✳ ❉♦r❛✈❛♥t❡✱ ❡ss❛s ❝✐t❛çõ❡s s❡rã♦ ❝❤❛♠❛❞❛s ❞❡ ❇✐❜❧✐♦❣r❛✜❛ ❇ás✐❝❛✳

✷✳✶ ❉❡✜♥✐çõ❡s ❜ás✐❝❛s

❉❡✜♥✐çã♦ ✷✳✶✳✶ ✭Φ✲á❧❣❡❜r❛✮✳ ❙❡❥❛Φ✉♠ ❛♥❡❧ ❛ss♦❝✐❛t✐✈♦✱ ❝♦♠✉t❛t✐✈♦ ❡ ✉♥✐tár✐♦✳ ❉✐r❡♠♦s q✉❡ ✉♠ Φ✲♠ó❞✉❧♦ ✉♥✐tár✐♦ A é ✉♠❛ á❧❣❡❜r❛ s❡ ❤♦✉✈❡r ✉♠❛ ♠✉❧t✐♣❧✐❝❛çã♦ ✏·✑ ❞❡✜♥✐❞❛ s♦❜r❡ A q✉❡ s❛t✐s❢❛ç❛ ♦s ❝r✐tér✐♦s ❛❜❛✐①♦✿

A1) (a+b)c=ac+bc✱ ♣❛r❛ q✉❛✐sq✉❡r a, b, c_∈A❀

A2) c(a+b) =ca+cb✱ ♣❛r❛ q✉❛✐sq✉❡r a, b, c_∈A❀

A1) γ(ab) = (γa)b =a(γb)✱ ♣❛r❛ q✉❛✐sq✉❡r a, b_∈A ❡ γ _∈Φ✳

❆ ♠❡♥♦s q✉❡ s❡ ❡s♣❡❝✐✜q✉❡ ♦ ❝♦♥trár✐♦✱ Φ s❡♠♣r❡ ❞❡♥♦t❛rá ✉♠ ❛♥❡❧ ❛ss♦❝✐❛t✐✈♦✱ ❝♦✲ ♠✉t❛t✐✈♦ ❡ ✉♥✐tár✐♦✱ ❝♦♥❤❡❝✐❞♦ t❛♠❜é♠ ❝♦♠♦ ❛♥❡❧ ❞❡ ♦♣❡r❛❞♦r❡s ❞❛ á❧❣❡❜r❛ A✳

❉❡✜♥✐çã♦ ✷✳✶✳✷ ✭❈❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✮✳ ❙❡❥❛♠I ✉♠ ✐❞❡❛❧ ❞❡ ✉♠ ❛♥❡❧R❡x, y❡❧❡♠❡♥t♦s ❞❡ R✳ ❉✐③❡♠♦s q✉❡ x é ❡q✉✐✈❛❧❡♥t❡ ❛ y ✭♠ó❞✉❧♦ I✮ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x₋y _∈I✳ ◆♦t❛çã♦✿ x_≡y✳ ■ss♦ ❞❡✜♥❡ ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♠R❡ ❞❡♥♦t❛r❡♠♦s ❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❛ q✉❛❧ x ❢❛③ ♣❛rt❡ ♣♦r

x+I = ¯x=_{x+a; a_∈I_}.

P♦❞❡♠♦s ❞❡✜♥✐r ♦♣❡r❛çõ❡s ✭✐♥❞✉③✐❞❛s✮ ♥♦ ❝♦♥❥✉♥t♦ ❞❛s ❝❧❛ss❡s ❛ ♣❛rt✐r ❞❛s ♦♣❡r❛çõ❡s ❞♦ ❛♥❡❧ R✱ ❝♦♠♦ ❛ s❡❣✉✐r✿

✶✳ (x+I) + (y+I) = (x+y) +I ✷✳ (x+I)(y+I) = xy+I

❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❝❧❛ss❡s s✉♣r❛❝✐t❛❞❛s ♠✉♥✐❞♦ ❞❛s ♦♣❡r❛çõ❡s ❛❝✐♠❛ ❞❡✜♥✐❞❛s s❡rá ❝❤❛♠❛❞♦ ❞❡ ❛♥❡❧ q✉♦❝✐❡♥t❡ ♠ó❞✉❧♦ I ❡ s❡rá ❞❡♥♦t❛❞♦ ♣♦r R/I✳

❉❡✜♥✐çã♦ ✷✳✶✳✸ ✭Pr♦❞✉t♦ ❞❡ ✐❞❡❛✐s✮✳ ❙❡ I ❡ J sã♦ ✐❞❡❛✐s ❞❡ ✉♠❛ á❧❣❡❜r❛ A✱ ❡♥tã♦ ❞❡✜♥✐♠♦s ♦ ♣r♦❞✉t♦ IJ ❡♥tr❡ ♦s ✐❞❡❛✐s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛

IJ = (

X

k

akbk;ak∈I , bk∈J

) .

(16)

✹ ❊▲❊▼❊◆❚❖❙ ●❊❘❆■❙ ❊▼ ➪▲●❊❇❘❆❙ ✷✳✶

❙❡❥❛ R ✉♠ ❛♥❡❧ ❡ ❝♦♥s✐❞❡r❡♠♦s ♦ ❝♦♥❥✉♥t♦ Mn(R) ❝♦♠♦ ❛ s❡❣✉✐r✿

Mn(R) =

             

a11 a12 · · · a1n

a21 a22 _{· · ·} a2n

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ an1 an2 · · · ann

    

; aij ∈A, 1≤i≤n, 1≤j ≤n

         .

❖ ❝♦♥❥✉♥t♦ Mn(R) ♠✉♥✐❞♦s ❞❛s ♦♣❡r❛çõ❡s ✏✰✑ ❡ ✏·✑ ✉s✉❛✐s ♣❛r❛ ♠❛tr✐③❡s é ✉♠ ❛♥❡❧✱

❝♦♥❤❡❝✐❞♦ ♣♦r ❛♥❡❧ ❞❡ ♠❛tr✐③❡s q✉❛❞r❛❞❛s ❞❡ ♦r❞❡♠ n s♦❜r❡ ♦ ❛♥❡❧ R ♦✉✱ s✐♠♣❧❡s♠❡♥t❡✱ ❛♥❡❧ ❞❡ ♠❛tr✐③❡s ❞❡ ♦r❞❡♠ n s♦❜r❡ ♦ ❛♥❡❧ R✳

❊①❡♠♣❧♦ ✷✳✶✳✹ ✭❆♥é✐s✮✳ ❙❡❥❛R✉♠ ❛♥❡❧ ❡✱ ❞❛❞♦s ✉♠ ✐♥t❡✐r♦k_∈Z❡ ✉♠ ❡❧❡♠❡♥t♦a_∈R✱ ❞❡✜♥❛ q✉❡ s♦❜r❡ R ❛ s❡❣✉✐♥t❡ ♦♣❡r❛çã♦✳

ka=a+_{· · ·}+a | {z }

k✲✈❡③❡s

.

❈♦♠ ❡st❛ ❞❡✜♥✐çã♦✱ ❛❞♦t❛r❡♠♦s t❛♠❜é♠ ❛s s❡❣✉✐♥t❡s ❝♦♥✈❡♥çõ❡s✿

0 |{z}

∈Z

a= 0 |{z}

∈R

, 1 |{z}

∈Z

a=a ❡ (₋1) | {z }

∈Z

a= ₋a |{z}

∈R

.

❉❡st❛ ❢♦r♠❛✱ q✉❛❧q✉❡r ❛♥❡❧ R é ✉♠❛Z✲á❧❣❡❜r❛✳

❊①❡♠♣❧♦ ✷✳✶✳✺ ✭❩❡r♦ á❧❣❡❜r❛s✮✳ ❯♠❛ Φ✲á❧❣❡❜r❛ A é ✉♠❛ ③❡r♦ á❧❣❡❜r❛ s❡ab= 0✱ q✉❛✐s✲ q✉❡r q✉❡ s❡❥❛♠ a, b_∈A✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ✉♠❛ ③❡r♦ á❧❣❡❜r❛ é ✉♠❛ á❧❣❡❜r❛ ♠✉♥✐❞❛ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ tr✐✈✐❛❧✳

❊①❡♠♣❧♦ ✷✳✶✳✻ ✭➪❧❣❡❜r❛ ❞❡ ●r❛ss♠❛♥♥✮✳ ❙❡❥❛♠K ✉♠ ❝♦r♣♦ ❡G✉♠K✲❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❉❡✜♥❛ ❡♠ G✉♠❛ ♠✉❧t✐♣❧✐❝❛çã♦ ✏·✑ ❛ss♦❝✐❛t✐✈❛✱ q✉❡ ❛❞♠✐t❡ ❡❧❡♠❡♥t♦ ✐❞❡♥t✐❞❛❞❡ ✶ ❡ q✉❡✱ ♣❛r❛ ♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠❛ ❜❛s❡ {uλ}λ∈Λ ✭❜❛s❡ ❞❡ G ❝♦♠♦ K✲❡s♣❛ç♦ ✈❡t♦r✐❛❧✮✱ uλ 6= 1✱ ❛s

s❡❣✉✐♥t❡s r❡❧❛çõ❡s s❡❥❛♠ ✈á❧✐❞❛s✿

u2_λ = 0;

uλiuλj =−uλjuλi, s❡ λi 6=λj.

❉❡✜♥✐♠♦s ❛ á❧❣❡❜r❛ ❞❡ ●r❛ss♠❛♥♥✱ ❞❡♥♦t❛❞❛ ♣♦r G✱ ❝♦♠♦ s❡♥❞♦ ❛ á❧❣❡❜r❛ ❣❡r❛❞❛ ♣❡❧♦s ❡❧❡♠❡♥t♦s {1_{} ∪ {}uλ, λ∈Λ}✱ ✐st♦ é✱

G= alg_h1,_{uλ}λ∈Λ, u2λ = 0, uλiuλj =−uλjuλi, s❡λi 6=λji.

❙❡ Λ =_{1,_{· · ·} , n_}✱ ❞❡♥♦t❛r❡♠♦s t❛❧ á❧❣❡❜r❛ ❞❡ ●r❛ss♠❛♥♥ ♣♦r Gn✳

❊①❡♠♣❧♦ ✷✳✶✳✼ ✭➪❧❣❡❜r❛s ❝♦♠✉t❛t✐✈❛s✮✳ ❯♠❛ á❧❣❡❜r❛ A é ❞✐t❛ s❡r ❝♦♠✉t❛t✐✈❛ s❡✱ ♣❛r❛ q✉❛✐sq✉❡r ❡❧❡♠❡♥t♦s a, b_∈A✱ é ❡q✉❛çã♦ ❛❜❛✐①♦ é ✈❡r❞❛❞❡✐r❛✿

ab₋ba= 0. ✭✷✳✶✮

❊①❡♠♣❧♦ ✷✳✶✳✽ ✭➪❧❣❡❜r❛s ❛♥t✐❝♦♠✉t❛t✐✈❛s✮✳ ❯♠❛ á❧❣❡❜r❛ A é ❞✐t❛ s❡r ❛♥t✐❝♦♠✉t❛t✐✈❛ s❡✱ ♣❛r❛ q✉❛✐sq✉❡r ❡❧❡♠❡♥t♦s a, b_∈A✱ é ❡q✉❛çã♦ ❛❜❛✐①♦ é ✈❡r❞❛❞❡✐r❛✿

(17)

✷✳✶ ❉❊❋■◆■➬Õ❊❙ ❇➪❙■❈❆❙ ✺

❊①❡♠♣❧♦ ✷✳✶✳✾ ✭➪❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s✮✳ ❯♠❛ á❧❣❡❜r❛ A é ❞✐t❛ s❡r ❛ss♦❝✐❛t✐✈❛ s❡✱ ♣❛r❛ q✉❛✐sq✉❡r ❡❧❡♠❡♥t♦s a, b, c_∈A✱ é ❡q✉❛çã♦ ❛❜❛✐①♦ é ✈❡r❞❛❞❡✐r❛✿

(ab)c₋a(bc) = 0. ✭✷✳✸✮

❊①❡♠♣❧♦ ✷✳✶✳✶✵ ✭➪❧❣❡❜r❛s ❞❡ ♣♦tê♥❝✐❛s ❛ss♦❝✐❛t✐✈❛s✮✳ ❯♠❛ á❧❣❡❜r❛ é ❞✐t❛ s❡r ❞❡ ♣♦tê♥✲ ❝✐❛s ❛ss♦❝✐❛t✐✈❛s s❡ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❣❡r❛ ✉♠❛ s✉❜á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛✳

❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ s❡Aé ❞❡ ♣♦tê♥❝✐❛s ❛ss♦❝✐❛t✐✈❛s ❡a_∈Aé ✉♠ ❡❧❡♠❡♥t♦ ❛r❜✐trár✐♦✱ ❡♥tã♦ q✉❛❧q✉❡r ❛rr❛♥❥♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ❝♦♠ r❡♣❡t✐çõ❡s ❞♦ ❡❧❡♠❡♥t♦s a ♥ã♦ ❞❡♣❡♥❞❡ ❞❛s ♣♦ssí✈❡✐s ❛❧♦❝❛çõ❡s ❡♥tr❡ ♦s ♣❛rê♥t❡s❡s✳

➪❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s sã♦✱ ❡✈✐❞❡♥t❡♠❡♥t❡✱ á❧❣❡❜r❛s ❞❡ ♣♦tê♥❝✐❛s ❛ss♦❝✐❛t✐✈❛s✳ ❚♦❞❛✈✐❛✱ ❛❧❣✉♥s ❡①❡♠♣❧♦s ♥ã♦ tr✐✈✐❛✐s ❞❡ á❧❣❡❜r❛s ❞❡ ♣♦tê♥❝✐❛s ❛ss♦❝✐❛t✐✈❛s s❡rã♦ ❝♦♥s✐❞❡r❛❞♦s ❛ s❡❣✉✐r✳ ❖ ✏♥ã♦ tr✐✈✐❛❧✑ ❛í ❞❡✈❡ s❡r ❡♥t❡♥❞✐❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ á❧❣❡❜r❛s q✉❡ ♥ã♦ sã♦ ✭♥❡❝❡ss❛r✐❛♠❡♥t❡✮ ❛ss♦❝✐❛t✐✈❛s✱ ♠❛s sã♦ ❞❡ ♣♦tê♥❝✐❛s ❛ss♦❝✐❛t✐✈❛s✳

❊①❡♠♣❧♦ ✷✳✶✳✶✶ ✭➪❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥✮✳ ❯♠❛ á❧❣❡❜r❛ A é ❞✐t❛ s❡r ❞❡ ❏♦r❞❛♥ s❡✱ ♣❛r❛ q✉❛✐sq✉❡r ❡❧❡♠❡♥t♦s a, b_∈A✱ ❛s ❞✉❛s r❡❧❛çõ❡s ❛❜❛✐①♦ sã♦ ✈á❧✐❞❛s✿

ab₋ba = 0; ✭✷✳✹✮

a2(ba)₋(a2b)a = 0. ✭✷✳✺✮

◆♦t❡ q✉❡ ❛s r❡❧❛çõ❡s ✷✳✶ ❡ ✷✳✹ ❝♦✐♥❝✐❞❡♠✱ ♦✉ s❡❥❛✱ á❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥ sã♦ á❧❣❡❜r❛s ❝♦♠✉t❛t✐✈❛s q✉❡ s❛t✐s❢❛③❡♠ ✷✳✺✳

❊①❡♠♣❧♦ ✷✳✶✳✶✷ ✭➪❧❣❡❜r❛s ❛❧t❡r♥❛t✐✈❛s✮✳ ❯♠❛ á❧❣❡❜r❛A é ❞✐t❛ s❡r ❛❧t❡r♥❛t✐✈❛ s❡✱ ♣❛r❛ q✉❛✐sq✉❡r ❡❧❡♠❡♥t♦s a, b_∈A✱ ❛s ❞✉❛s r❡❧❛çõ❡s ❛❜❛✐①♦ sã♦ ✈á❧✐❞❛s✿

(ab)b₋ab2 = 0; ✭✷✳✻✮

a2b₋a(ab) = 0. ✭✷✳✼✮

❆ r❡❧❛çã♦✷✳✻ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❛❧t❡r♥❛t✐✈✐❞❛❞❡ à ❞✐r❡✐t❛✳ P♦r s✉❛ ✈❡③✱ ❛ r❡❧❛çã♦ ✷✳✼ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❛❧t❡r♥❛t✐✈✐❞❛❞❡ à ❡sq✉❡r❞❛✳ P♦rt❛♥t♦✱ á❧❣❡❜r❛s ❛❧t❡r♥❛t✐✈❛s sã♦ á❧❣❡❜r❛s ❛❧t❡r♥❛t✐✈❛s à ❞✐r❡✐t❛ ❡ à ❡sq✉❡r❞❛✳

❊①❡♠♣❧♦ ✷✳✶✳✶✸ ✭➪❧❣❡❜r❛s ✢❡①í✈❡✐s✮✳ ❯♠❛ á❧❣❡❜r❛Aé ❞✐t❛ s❡r ✢❡①í✈❡❧ s❡✱ ♣❛r❛ q✉❛✐sq✉❡r ❡❧❡♠❡♥t♦s a, b_∈A✱ é ❡q✉❛çã♦ ❛❜❛✐①♦ é ✈❡r❞❛❞❡✐r❛✿

a(ba)₋(ab)a = 0. ✭✷✳✽✮

❙❡❥❛ A ✉♠❛ á❧❣❡❜r❛ ❝♦♠✉t❛t✐✈❛ ✭❊①❡♠♣❧♦ ✷✳✶✳✼✮ ❡ s❡❥❛♠ a ❡ b ❡❧❡♠❡♥t♦s ❛r❜✐trár✐♦s ❞❡ A✳ ❖❜s❡r✈❡ q✉❡

(ab)a₋a(ba) = |{z} ❝♦♠✉t❛t✐✈✐❞❛❞❡

(ab)a₋(ba)a = |{z} ❝♦♠✉t❛t✐✈✐❞❛❞❡

(ab)a₋(ab)a= 0.

(18)

✻ ❊▲❊▼❊◆❚❖❙ ●❊❘❆■❙ ❊▼ ➪▲●❊❇❘❆❙ ✷✳✶

❱❡r❡♠♦s✱ ♥♦ ❈❛♣ít✉❧♦✺✭s♦❜r❡ á❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥ ♥ã♦ ❝♦♠✉t❛t✐✈❛s✮✱ q✉❡ ❛ ✢❡①✐❜✐❧✐❞❛❞❡ ♣♦❞❡ s❡r ❡♥t❡♥❞✐❞❛ ❝♦♠♦ ✉♠❛ ❝♦♥❞✐çã♦ ♠❛✐s ❢r❛❝❛ à ❝♦♠✉t❛t✐✈✐❞❛❞❡✳

❊①❡♠♣❧♦ ✷✳✶✳✶✹ ✭➪❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥ ♥ã♦ ❝♦♠✉t❛t✐✈❛s✮✳ ❯♠❛ á❧❣❡❜r❛ A é ❞✐t❛ s❡r ❞❡ ❏♦r❞❛♥ ♥ã♦ ❝♦♠✉t❛t✐✈❛ s❡✱ ♣❛r❛ q✉❛✐sq✉❡r ❡❧❡♠❡♥t♦sa, b_∈A✱ ❛s ❞✉❛s r❡❧❛çõ❡s ❛❜❛✐①♦ sã♦ ✈á❧✐❞❛s✿

a(ba)₋(ab)a = 0. ✭✷✳✾✮

a2(ba)₋(a2b)a = 0. ✭✷✳✶✵✮ ❆ r❡❧❛çã♦✷✳✾♠♦str❛✲♥♦s q✉❡ á❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥ ♥ã♦ ❝♦♠✉t❛t✐✈❛s sã♦ á❧❣❡❜r❛s ✢❡①í✈❡✐s q✉❡ s❛t✐s❢❛③❡♠ ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ❏♦r❞❛♥✷✳✺✳

❊♠ ❬❩❙❙❙✽✷❪✱ ♣♦❞❡✲s❡ ✈❡r✐✜❝❛r q✉❡ á❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥✱ á❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥ ♥ã♦ ❝♦✲ ♠✉t❛t✐✈❛s ❡ á❧❣❡❜r❛s ❛❧t❡r♥❛t✐✈❛s sã♦ á❧❣❡❜r❛s ❞❡ ♣♦tê♥❝✐❛s ❛ss♦❝✐❛t✐✈❛s✳ ❊ss❛ ✐♥❢♦r♠❛çã♦ ♥♦s s❡rá út✐❧ ♠❛✐s ❛❞✐❛♥t❡✳

❉❛❞♦s ✉♠ ❝♦♥❥✉♥t♦X ♠✉♥✐❞♦ ❞❡ ✉♠❛ ♦♣❡r❛çã♦ ✏_·✑ ❡ ✜①❛❞♦ ✉♠ ❡❧❡♠❡♥t♦a _∈X✱ ❞❡✜✲ ♥✐r❡♠♦s ❛s ❢✉♥çõ❡s ♠✉❧t✐♣❧✐❝❛çã♦ à ❞✐r❡✐t❛✱ ❞❡♥♦t❛❞❛ ♣♦r Ra✱ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ à ❡sq✉❡r❞❛✱

❞❡♥♦t❛❞❛ ♣♦r La✱ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

Ra :X → X

x _7→ x_·a;

La:X → X

x _7→ a_·x.

❆❞♦t❛r❡♠♦s ❛ ♥♦t❛çã♦ xRa = xa ♣❛r❛ ✐♥❞✐❝❛r ❛ ✐♠❛❣❡♠ ❞♦ ❡❧❡♠❡♥t♦ ❛tr❛✈és ❞❛

❛♣❧✐❝❛çã♦ Ra✳ ❆❧❣✉♠❛s ♦❜r❛s ✉s❛♠ ❛ ♥♦t❛çã♦ ✉s✉❛❧ ♣❛r❛ ❢✉♥çõ❡s✱ Ra(x) = xa✱ ♠❛s ✐ss♦

s❡rá ❡✈✐t❛❞♦ ♥❡st❡ tr❛❜❛❧❤♦✱ ❛♣❡♥❛s ♣♦r ❝♦♥✈❡♥çã♦ ❞❛ ❧✐t❡r❛t✉r❛ ❛t✉❛❧✳ ❖ ♠❡s♠♦ s❡ ❛♣❧✐❝❛ ❛ La✳

❖s ❡①❡♠♣❧♦s ❛♥t❡r✐♦r❡s ❥á s✉❣❡r❡♠ q✉❡✱ ♥❡ss❡ tr❛❜❛❧❤♦✱ ♥❡♠ s❡♠♣r❡ ❛s ♦♣❡r❛çõ❡s s❡rã♦ ❛ss♦❝✐❛t✐✈❛s ♦✉ ❝♦♠✉t❛t✐✈❛s✳ ❉❡ss❛ ❢♦r♠❛✱ ✐♥tr♦❞✉③✐r❡♠♦s ❞♦✐s ♦♣❡r❛❞♦r❡s✿ ♦ ❛ss♦❝✐❛❞♦r ❡ ♦ ❝♦♠✉t❛❞♦r✳

❉❡✜♥✐çã♦ ✷✳✶✳✶✺ ✭❆ss♦❝✐❛❞♦r✮✳ ❙❡❥❛♠X ✉♠ ❝♦♥❥✉♥t♦ ❡_·✉♠❛ ♦♣❡r❛çã♦ s♦❜r❡X✳ ❉❛❞♦s x, y ❡ z ❡❧❡♠❡♥t♦s ❛r❜✐trár✐♦s ❡♠ X✱ ❞❡✜♥✐r❡♠♦s ♦ ❛ss♦❝✐❛❞♦r ♣❡❧❛ s❡❣✉✐♥t❡ ❡①♣r❡ssã♦✿

(x, y, z) = (x_·y)_·z₋x_·(y_·z).

❉❡✜♥✐çã♦ ✷✳✶✳✶✻ ✭❈♦♠✉t❛❞♦r✮✳ ❙❡❥❛♠X ✉♠ ❝♦♥❥✉♥t♦ ❡_·✉♠❛ ♦♣❡r❛çã♦ s♦❜r❡X✳ ❉❛❞♦s x ❡y ❡❧❡♠❡♥t♦s ❛r❜✐trár✐♦s ❡♠ X✱ ❞❡✜♥✐r❡♠♦s ♦ ❝♦♠✉t❛❞♦r ❛tr❛✈és ❞❛ ❡①♣r❡ssã♦ ❛❜❛✐①♦✿

[x, y] =x_·y₋y_·x.

(19)

✷✳✶ ❉❊❋■◆■➬Õ❊❙ ❇➪❙■❈❆❙ ✼

♣❛rt✐r ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❥á ❞❡✜♥✐❞❛ ❡♠ A✱ ❝♦♠♦ ❛ s❡❣✉✐r✿

a_◦b = 1

2(ab+ba).

❙❡❣✉❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ✏◦✑ q✉❡a_◦b=b_◦a✱ ♦✉ s❡❥❛✱(A,+,_◦)é ✉♠❛ á❧❣❡❜r❛ ❝♦♠✉t❛t✐✈❛✳ ❉❡✜♥✐♠♦s✱ ♣♦rt❛♥t♦✱ ❛ á❧❣❡❜r❛ ❛❞❥✉♥t❛ A(+) ❝♦♠♦ s❡♥❞♦ A(+) = (A,+,_◦) ❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ✏◦✑ t❛♠❜é♠ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❏♦r❞❛♥✳

❊♠A(+) ♣♦❞❡♠♦s t❛♠❜é♠ ❞❡✜♥✐r ✉♠ ❛ss♦❝✐❛❞♦r r❡❧❛❝✐♦♥❛❞♦ à ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❏♦r✲ ❞❛♥✱ ❛ q✉❡ ❝❤❛♠❛r❡♠♦s ❞❡ ❛ss♦❝✐❛❞♦r+ ❡ ❞❡♥♦t❛r❡♠♦s ♣♦r(a, b, c)+✿

(a, b, c)+ := (a_◦b)_◦c₋a_◦(b_◦c), _∀a, b, c_∈A.

❊♠ ❬❩❙❙❙✽✷✱ ♣✳✺✸❪ ❡♥❝♦♥tr❛♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿

▲❡♠❛ ✷✳✶✳✶✼✳ P❛r❛ q✉❛✐sq✉❡r ❡❧❡♠❡♥t♦s a, b, c_∈A✱ ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦ é ✈á❧✐❞❛✿

4(a, b, c)+ = (a, b, c)₋(c, b, a) + (b, a, c)₋(c, a, b) + (a, c, b)₋(b, c, a) + [b,[a, c]],

❡♠ q✉❡ (a, b, c)+ é ♦ ❛ss♦❝✐❛❞♦r ❞❡✜♥✐❞♦ ❡♠ A(+)= (A,+,_◦)✳ ❙❡❥❛A ✉♠❛ á❧❣❡❜r❛✳ ❈♦♥s✐❞❡r❡ ❡♠ A ♦s s❡❣✉✐♥t❡s s✉❜❝♦♥❥✉♥t♦s✿

N(A) = _{n_∈A|(n, A, A) = (A, n, A) = (A, A, n) = 0_}; K(A) = _{k_∈A|[k, A] = 0_};

Z(A) = N(A)_∩K(A).

❖s ❝♦♥❥✉♥t♦sN(A), K(A)❡Z(A)sã♦ ❝♦♥❤❡❝✐❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❝♦♠♦ ❝❡♥tr♦ ❛ss♦✲ ❝✐❛t✐✈♦✱ ❝❡♥tr♦ ❝♦♠✉t❛t✐✈♦ ❡ ❝❡♥tr♦ ❞❛ á❧❣❡❜r❛ A❡ sã♦ ♦s ♣r✐♥❝✐♣❛✐s s✉❜❝♦♥❥✉♥t♦s ❝❡♥tr❛✐s ❞❡ ✉♠❛ á❧❣❡❜r❛ A✳

❙❡❥❛♠ A ✉♠❛ á❧❣❡❜r❛ ❡ Z = Z(A) s❡✉ ❝❡♥tr♦✳ ❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛ à ❝♦♥str✉çã♦ ❞♦ ❝♦r♣♦ ❞♦s r❛❝✐♦♥❛✐s Q ❛ ♣❛rt✐r ❞❡ Z×Z∗_{✱ ❞❡✜♥❛ ♦ s✐st❡♠❛ ♠✉❧t✐♣❧✐❝❛t✐✈♦} _Z−1 ₌_Z_{\ {}₀_} ❡ ❝♦♥s✐❞❡r❡ ❛ á❧❣❡❜r❛ Z−1_{A✳ ❙❡} _B ∼₌_Z−1_{A✱ ❡♥tã♦ ❞✐r❡♠♦s q✉❡} _B _{é ♦ ❢❡❝❤♦ ❝❡♥tr❛❧ ❞❡} _A ♦✉ A é ✉♠❛ ♦r❞❡♠ ❝❡♥tr❛❧ ❡♠B✳

❙❡ F é ✉♠ ❝♦r♣♦ ❡ A é ✉♠❛ F✲á❧❣❡❜r❛ t❛❧ q✉❡ Z(A) = Z = F✱ ❡♥tã♦ A é ❞✐t❛ s❡r ❝❡♥tr❛❧ s♦❜r❡ F✳ ❯♠ á❧❣❡❜r❛ A é ❞✐t❛ s❡r ❝❡♥tr❛❧ s✐♠♣❧❡s s❡ ❡❧❛ é ❝❡♥tr❛❧ ❡ s✐♠♣❧❡s ✭❛ s✐♠♣❧✐❝✐❞❛❞❡ ❞❡ ✉♠❛ á❧❣❡❜r❛ s❡❣✉❡✲s❡ ♥♦ s❡♥t✐❞♦ ✉s✉❛❧ ❞❛ ♣❛❧❛✈r❛✿ ✉♠❛ á❧❣❡❜r❛B ₆= (0)é s✐♠♣❧❡s s❡ ♦s ú♥✐❝♦s ✐❞❡❛✐s ❜✐❧❛t❡r❛✐s ❞❡Bsã♦(0)❡B✮✳ ❙❡ ✉♠❛ á❧❣❡❜r❛Aé ❝❡♥tr❛❧ s✐♠♣❧❡s ❝♦♠ ❡❧❡♠❡♥t♦ ✐❞❡♥t✐❞❛❞❡ ✶✱ ❡♥tã♦ Z(A) é ✐s♦♠♦r❢♦ ❛♦ ❝♦r♣♦ F ❛tr❛✈és ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r✿ Z(A)∼=F _·1∼=F✳

❉❡✜♥✐çã♦ ✷✳✶✳✶ ✭❙♦♠❛ ❞✐r❡t❛✮✳ ❙❡❥❛ A ✉♠❛ á❧❣❡❜r❛ ❡ s❡❥❛♠I1, I2, ..., In ✐❞❡❛✐s ❞❡ A✳ ❙❡

A = Pn_j₌₁Ij ❡ Ij ∩

X

1≤l≤n;l6=j

Il

!

= (0)✱ ❡♥tã♦ ❞✐r❡♠♦s q✉❡ A é ✉♠❛ s♦♠❛ ❞✐r❡t❛ ❞♦s

✐❞❡❛✐s I1, I2, . . . , In✳ ◆♦t❛çã♦✿ A=I1⊕I2⊕ · · ·In= n

M

j=1 Ij .

(20)

✽ ❊▲❊▼❊◆❚❖❙ ●❊❘❆■❙ ❊▼ ➪▲●❊❇❘❆❙ ✷✳✶

❞❡ á❧❣❡❜r❛s ✐♥❞❡①❛❞❛s ♣♦r ✉♠ ❝♦♥❥✉♥t♦ Λ ✭❞❡ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❛r❜✐trár✐❛✮✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ❝♦♥❥✉♥t♦ S =Q_λ_∈_ΛAλ ✭✏Q✑ ❞❡♥♦t❛ ♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦✮ ♠✉♥✐❞♦ ❞❛s ♦♣❡r❛çõ❡s ❛❞✐çã♦

❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡✜♥✐❞❛s ❝♦♦r❞❡♥❛❞❛ ❛ ❝♦♦r❞❡♥❛❞❛✱ ✐st♦ é✱

(aλ)λ+ (bλ)λ = (aλ+bλ)λ

❡

(aλ)λ ·(bλ)λ = (aλ·bλ)λ.

❉❡✜♥✐❞♦ ❞❡ss❛ ❢♦r♠❛ ✭❛s ♦♣❡r❛çõ❡s ❡♠ q✉❡stã♦ ❡stã♦ ❜❡♠ ❞❡✜♥✐❞❛s✮✱Sé ❞✐t♦ s❡r ♦ ♣r♦❞✉t♦ ❞✐r❡t♦ ❞❛s á❧❣❡❜r❛sAλ✱λ∈Λ✱ ❡ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞❛ ❝♦♠♦ ✉♠❛ ❡①t❡♥sã♦ ❞❛ s♦♠❛ ❞✐r❡t❛

♣♦✐s ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❛ ❢♦r♠❛

aα, s❡ λ=α

0, s❡ λ₆=α := (0,· · · ,0, aα,0,· · · ,0), ♥❛α✲és✐♠❛ ♣♦s✐çã♦✱ é ✉♠ ✐❞❡❛❧ A′α ❞❡S q✉❡ é ✐s♦♠♦r❢♦ ❛ Aα ❡ ❛ ❛♣❧✐❝❛çã♦

S =Q_λ_∈_ΛAλ → A′α

(aλ)λ 7→ (0,· · · ,0, aα,0,· · · ,0) (α✲és✐♠❛ ♣♦s✐çã♦)

é ✉♠ ❡♣✐♠♦r✜s♠♦ ❞❡ S ❡♠ A′

α✳

❆ á❧❣❡❜r❛ Sw ❞❡ S q✉❡ ❝♦♥s✐st❡ ❞❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s q✉❡ ♣♦ss✉❡♠ ✉♠❛ q✉❛♥t✐❞❛❞❡

✜♥✐t❛ ❞❡ ❡♥tr❛❞❛s ♥ã♦ ♥✉❧❛s é ❝❤❛♠❛❞♦ ❞❡ s♦♠❛ ❞✐r❡t❛ ❞❛ ❝♦❧❡çã♦ {Aλ}λ∈Λ✳ ◆♦t❡ q✉❡ ♦

❤♦♠♦♠♦r✜s♠♦ ♥❛t✉r❛❧ ❞❡✜♥✐❞♦ ❛❝✐♠❛✱ q✉❛♥❞♦ r❡str✐t♦ ❛Sw✱ é ✉♠❛ s♦❜r❡❥❡çã♦ ❞❡ Sw ❡♠

A′

α✱ q✉❛❧q✉❡r q✉❡ s❡❥❛ α ∈ Λ✳ ❉✐r❡♠♦s q✉❡ ❛ s✉❜á❧❣❡❜r❛ S∗ ❞❡ S é ✉♠❛ s♦♠❛ s✉❜❞✐r❡t❛

❞❛ ❝♦❧❡çã♦ {Aλ}λ∈Λ s❡ ♦ ❤♦♠♦♠♦r✜s♠♦ ♥❛t✉r❛❧ ❞❡ S∗ ❡♠ A′α✱

(aλ)λ 7→ (0· · ·0, aα,0· · ·0),

é s♦❜r❡❥❡t✐✈♦✱ ♣❛r❛ t♦❞♦ α _∈ Λ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❡st❛♠♦s ❞✐③❡♥❞♦ q✉❡ t♦❞❛ s♦♠❛ ❞✐r❡t❛ ❝♦♠♣❧❡t❛ é ✉♠❛ s♦♠❛ s✉❜❞✐r❡t❛✳

❙❡ ❛ á❧❣❡❜r❛ A é ✉♠❛ s♦♠❛ s✉❜❞✐r❡t❛ ❞❛ ❝♦❧❡çã♦ {Aλ}λ∈Λ✱ ❡♥tã♦ ✉s❛r❡♠♦s ♦ sí♠❜♦❧♦

✏֒→✑ ♣❛r❛ ❡①♣r❡ss❛r ❡ss❛ ❝♦♥❞✐çã♦✿

A֒_→ Y

λ∈Λ Aλ.

❆♣r♦✈❡✐t❛♥❞♦ ❡st❡ ❡s♣❛ç♦✱ q✉❡ tr❛t❛ ❞❡ ♣r♦❞✉t♦s ❝❛rt❡s✐❛♥♦s✱ é út✐❧ r❡❧❡♠❜r❛r♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ t❡♥s♦r✳ ❙❡❥❛♠ R ✉♠ ❛♥❡❧✱ A ✉♠ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✱ B ✉♠ R✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛ ❡ G ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✱ ❞❡✜♥✐♠♦s ♦ ♣r♦❞✉t♦ ❜❛❧❛♥❝❡❛❞♦ P : A_×B _→G ❝♦♠♦ s❡♥❞♦ ✉♠❛ ❛♣❧✐❝❛çã♦ ❜✐❧✐♥❡❛r t❛❧ q✉❡

P(a_·r, b) =P(a, r_·b), _∀a_∈A, _∀b _∈B, _∀r_∈R.

❖ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ A_⊗RB é ❞❡✜♥✐❞♦ ✭✉♥✐❝❛♠❡♥t❡✮ ✈✐❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ✭♣❛r❛ ♦

♣r♦❞✉t♦ ❜❛❧❛♥❝❡❛❞♦ P✮ ❛tr❛✈és ❞❛ ❛♣❧✐❝❛çã♦

⊗: A_×B _→A_⊗RB.

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✷✳✷ ❱❆❘■❊❉❆❉❊❙ ✾

◆❡st❛ s❡çã♦✱ ❢♦r❛♠ ❡①♣♦st❛s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ❣❡r❛✐s ❛❝❡r❝❛ ❞❛ ❡str✉t✉r❛ ❞❛s á❧❣❡❜r❛s✳ ❆ s❡❣✉✐r✱ ♥♦s ♦❝✉♣❛r❡♠♦s ❡♠ ❢✉♥❞❛♠❡♥t❛r ♦❜❥❡t♦s q✉❡ tê♠ ♠❛✐♦r s✐❣♥✐✜❝â♥❝✐❛ ♣❛r❛ ♥♦ss♦ ❡st✉❞♦✳

✷✳✷ ❱❛r✐❡❞❛❞❡s

❱❛r✐❡❞❛❞❡s ❞❡ á❧❣❡❜r❛s sã♦ ❝❧❛ss❡s ❞❡ á❧❣❡❜r❛s ❞❡t❡r♠✐♥❛❞❛s ♣♦r ❝❡rt❛s ✐❞❡♥t✐❞❛❞❡s✳ ❋✐①❡ ✉♠ ❝♦♥❥✉♥t♦ ❛r❜✐trár✐♦X =_{xα}❡ ❛❞✐❝✐♦♥❡ ❛ ❡❧❡ ♠❛✐s ❞♦✐s sí♠❜♦❧♦s✱ ♣❛rê♥t❡s❡s

à ❡sq✉❡r❞❛ ✏✭✑ ❡ ♣❛rê♥t❡s❡s à ❞✐r❡✐t❛ ✏✮✑✱ ♣❛r❛ ♦❜t❡r ✉♠ ♥♦✈♦ ❝♦♥❥✉♥t♦X∗ ₌_X

∪ {(, )_}✳ ❈♦♥s✐❞❡r❡ t♦❞❛s ❛s ♣♦ssí✈❡✐s s❡q✉ê♥❝✐❛s ✜♥✐t❛s ❞❡ ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ X∗ _{q✉❡ ♥ã♦}

♣♦ss✉❛♠ tr✐✈✐❛❧✐❞❛❞❡s ♦✉ ♣♦s✐çõ❡s ✐♥❛❞❡q✉❛❞❛s ♣❛r❛ ♦s ♣❛rê♥t❡s❡s ✭♣♦r ❡①❡♠♣❧♦✱ ❞❡s♣r❡③❡ ❛s ♣❛❧❛✈r❛s ✏✭✮✑✱ ✏✮①✭✑✱ ✏②✭①③✭✑✱ ❡t❝✮✳ ❉✉❛s s❡q✉ê♥❝✐❛s ✜♥✐t❛sa1a2_{· · ·}am❡b1b2,· · ·bn✱ ❡♠ q✉❡

ai, bj ∈ X∗✱ sã♦ ❝♦♥s✐❞❡r❛❞❛s ✐❣✉❛✐s s❡ m =n ❡ ai =bi✱ ♣❛r❛ i = 1,2,· · ·, m✳ ❉❡✜♥✐♠♦s

✐♥❞✉t✐✈❛♠❡♥t❡ ✉♠ ❝♦♥❥✉♥t♦ V[X]❞❡ss❛s s❡q✉ê♥❝✐❛s ❞❡ ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ X∗_{✱ ♦ q✉❛❧}

❝❤❛♠❛r❡♠♦s ❞❡ ♣❛❧❛✈r❛s ♥ã♦ ❛ss♦❝✐❛t✐✈❛s ❞♦ ❝♦♥❥✉♥t♦ X✳ Pr✐♠❡✐r♦✱ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ X ♣❡rt❡♥❝❡♠ ❛♦ ❝♦♥❥✉♥t♦ V[X]✳ ❙❡❣✉♥❞♦✱ s❡ x1, x2 _∈X ❡ u, v _∈ V[X]_\X✱ ❡♥tã♦ ❛s s❡qüê♥❝✐❛sx1x2✱x1(u)✱(v)x2 ❡(u)(v)t❛♠❜é♠ ♣❡rt❡♥❝❡♠ ❛♦ ❝♦♥❥✉♥t♦V[X]✳ ◆❡♠ t♦❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ X∗ _{é ✉♠ ❡❧❡♠❡♥t♦ ❞❡} _V_{[X]✳ P♦r ❡①❡♠♣❧♦✱ ❛ s❡q✉ê♥❝✐❛}

(x1(x2x3))x4 é ✉♠❛ ♣❛❧❛✈r❛ ♥ã♦ ❛ss♦❝✐❛t✐✈❛ ❞♦ ❝♦♥❥✉♥t♦ X = _{x1, x2, x3, x4_}✱ ♠❛s ❛ s❡q✉ê♥❝✐❛ (x1(x2x3)x4) ♥ã♦ é✳ ❖ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ X q✉❡ ❛♣❛r❡❝❡♠ ❡♠ ✉♠❛ ♣❛❧❛✈r❛u é ❝❤❛♠❛❞♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ♣❛❧❛✈r❛ ♥ã♦ ❛ss♦❝✐❛t✐✈❛ u ❡ é ✐♥❞✐❝❛❞♦ ♣♦r d(u)✳

P♦❞❡♠♦s ❞❡✜♥✐r s♦❜r❡ V[X] ✉♠❛ ♠✉❧t✐♣❧✐❝❛çã♦✳ ❙❡❥❛♠ x1, x2 ∈X ❡ u, v ∈V[X]\X✳ ❉❡✜♥❛✿

x1_·x2 =x1x2;

x1_·u=x1(u);

v_·x2 = (v)x2;

u_·v = (u)(v).

❈♦♥s✐❞❡r❡ ❛❣♦r❛ ♦ Φ✲♠ó❞✉❧♦ ✉♥✐tár✐♦ ❡ ❧✐✈r❡✱ Φ[X]✱ s♦❜r❡ ♦ ❝♦♥❥✉♥t♦ V[X] ❡ ❡①t❡♥❞❛ ❛ ♦♣❡r❛çã♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡✜♥✐❞❛ ❡♠ V[X] ♣❛r❛ Φ[X]♣❡❧❛ r❡❣r❛

X

i

αiui

!

· X

j

βjuj

!

=X

i,j

αiβj(ui·uj),

❡♠ q✉❡ αi, βj ∈ Φ❡ ui, vj ∈V[X]✳ ❈♦♠ ❡ss❛ ❝♦♥str✉çã♦✱ ♦❜t❡♠♦s ❛ á❧❣❡❜r❛ Φ[X]✱ q✉❡ é

❝❤❛♠❛❞❛ ❞❡ á❧❣❡❜r❛ ❧✐✈r❡ s♦❜r❡ ♦ ❛♥❡❧ ❞❡ Φ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❣❡r❛❞♦r❡s X✳

❋✐①❡✱ ❛❣♦r❛✱ ✉♠ ❝♦♥❥✉♥t♦ ❡♥✉♠❡rá✈❡❧ ❞❡ sí♠❜♦❧♦s X = _{x1, x2,· · · }✳ ❙❡❥❛ f ✉♠ ❡❧❡✲ ♠❡♥t♦ ❛r❜✐trár✐♦ ❞❡ Φ[X]✳ ▲♦❣♦✱ ❞❡✈❡ ❛♣❛r❡❝❡r ❡♠ s❡✉ s✉♣♦rt❡ ❛♣❡♥❛s ✉♠ ♥ú♠❡r♦ ✜✲ ♥✐t♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ X✱ ♣♦r ❡①❡♠♣❧♦ x1, x2,_{· · ·} , xn. ◆❡st❡ ❝❛s♦✱ ✈❛♠♦s ❡s❝r❡✈❡r f =

f(x1, x2,_{· · ·}, xn)✳ ❙❡ A é ✉♠❛ Φ✲á❧❣❡❜r❛ ❡ a1, . . . , an sã♦ ❡❧❡♠❡♥t♦s ❛r❜✐trár✐♦s ❡♠ A✱ ❡♥✲

tã♦ é ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ ❡①✐st❡ ✉♠ ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦θ : Φ[X]_→A✱ t❛❧ q✉❡xi 7→ai✱

♣❛r❛ i = 1,_{· · ·} , n✱ ❡ ❛♣❧✐❝❛ ♦s ♦✉tr♦s ❡❧❡♠❡♥t♦s ❞❡ X ♥♦ ③❡r♦✳ ■r❡♠♦s ❞❡♥♦t❛r ❛ ✐♠❛✲ ❣❡♠ ❞♦ ❡❧❡♠❡♥t♦ f ♣❡❧♦ ❤♦♠♦♠♦r✜s♠♦ θ ♣♦r f(a1, a2,_{· · ·} , an) ❡ ❞✐③❡r q✉❡ ♦ ❡❧❡♠❡♥t♦

f(a1, a2, ..., an) é ♦❜t✐❞♦ ♣♦r s✉❜st✐t✉✐çã♦ ❞♦ ❡❧❡♠❡♥t♦s a1, a2,· · · , an ♥♦ ♣♦❧✐♥ô♠✐♦ ♥ã♦

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✶✵ ❊▲❊▼❊◆❚❖❙ ●❊❘❆■❙ ❊▼ ➪▲●❊❇❘❆❙ ✷✳✷

❉❡✜♥✐çã♦ ✷✳✷✳✶ ✭■❞❡♥t✐❞❛❞❡✮✳ ❯♠ ♣♦❧✐♥ô♠✐♦ ♥ã♦ ❛ss♦❝✐❛t✐✈♦ f = f(x1, x2,_{· · ·} , xn) ∈

Φ[X] é ❝❤❛♠❛❞♦ ❞❡ ✐❞❡♥t✐❞❛❞❡ ❞❛ á❧❣❡❜r❛ A✱ s❡ f(a1, a2,_{· · ·}, an) = 0 ♣❛r❛ q✉❛✐sq✉❡r

a1, a2,_{· · ·} , an ∈A✳ ❉✐③❡♠♦s t❛♠❜é♠ q✉❡A s❛t✐s❢❛③ ❛ ✐❞❡♥t✐❞❛❞❡ f ♦✉ q✉❡ ❛ ✐❞❡♥t✐❞❛❞❡f

é ✈á❧✐❞❛ ❡♠ A✳

❆ ❝♦❧❡çã♦ ❞❡ t♦❞❛s ❛s ✐❞❡♥t✐❞❛❞❡s ❞❡ ✉♠❛ ❝❡rt❛ á❧❣❡❜r❛ é ✉♠ ✐❞❡❛❧ ❞❛ á❧❣❡❜r❛Φ[X]✱ ❛ q✉❛❧ é ❝❤❛♠❛❞❛ ✐❞❡❛❧ ❞❛s ✐❞❡♥t✐❞❛❞❡s ✭T✲✐❞❡❛❧✮ ❞❛ á❧❣❡❜r❛ ❞❡ ❆ ❡ é ❞❡♥♦t❛❞♦ ♣♦rT(A)✳ ❆ ❝♦❧❡çã♦ ❞❡ t♦❞❛s ❛s ✐❞❡♥t✐❞❛❞❡s q✉❡ sã♦ s❛t✐s❢❡✐t❛s ♣♦r ❝❛❞❛ á❧❣❡❜r❛ ❞❡ ✉♠❛ ❝❡rt❛ ❝❧❛ss❡ Mt❛♠❜é♠ é ✉♠ ✐❞❡❛❧ ❡♠ Φ[X]✳ ➱ ❝❤❛♠❛❞❛ ❞❡ ✐❞❡❛❧ ❞❛s ✐❞❡♥t✐❞❛❞❡s ✭T✲✐❞❡❛❧✮ ❞❛ ❝❧❛ss❡ ❞❛s á❧❣❡❜r❛s M❡ é ❞❡♥♦t❛❞♦ ♣♦r T(_M)✳

❉❡✜♥✐çã♦ ✷✳✷✳✷ ✭❱❛r✐❡❞❛❞❡✮✳ ❙❡❥❛ I ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ Φ[X]✳ ❆ ❝❧❛ss❡ _M ❞❡ t♦❞❛s ❛s Φ✲á❧❣❡❜r❛s s❛t✐s❢❛③❡♥❞♦ ❝❛❞❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ I é ❝❤❛♠❛❞❛ ❞❡ ✈❛r✐❡❞❛❞❡ ❞❡ Φ✲á❧❣❡❜r❛s ❞❡✜♥✐❞❛ ♣❡❧♦ ❝♦♥❥✉♥t♦ ❞❡ ✐❞❡♥t✐❞❛❞❡s I✳ I é t❛♠❜é♠ ❝❤❛♠❛❞♦ ❞❡ ❝♦♥❥✉♥t♦ ❞❡ ❞❡✜♥✐çã♦ ❞❛s ✐❞❡♥t✐❞❛❞❡s ❞❛ ✈❛r✐❡❞❛❞❡ M✳

❯♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ✈❛r✐❡❞❛❞❡ ♣♦❞❡ s❡r ❞❛❞❛ ❛ s❡❣✉✐r✿

❚❡♦r❡♠❛ ✷✳✷✳✶ ✭❇✐r❦❤♦✛✮✳ ❯♠❛ ❝❧❛ss❡ ❞❡ Φ✲á❧❣❡❜r❛s _V é ✉♠❛ ✈❛r✐❡❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ s❛t✐s❢❡✐t❛s✿

✐✮ V é ❢❡❝❤❛❞❛ ♣♦r s✉❜á❧❣❡❜r❛s✱ ✐st♦ é✱ s❡ A _{∈ V} ❡ B _⊂ A é ✉♠❛ s✉❜á❧❣❡❜r❛✱ ❡♥tã♦ B _{∈ V}✳

✐✐✮ V é ❢❡❝❤❛❞❛ ♣♦r ♣r♦❞✉t♦s ❞✐r❡t♦s✱ ✐st♦ é✱ s❡ {Aα} ⊂ V✱ ❡♥tã♦ Q_αAα ∈ V✳

✐✐✐✮ V é ❢❡❝❤❛❞❛ ♣♦r ✐♠❛❣❡♥s ❤♦♠♦♠ór✜❝❛s✱ ✐st♦ é✱ s❡ A _{∈ V} ❡ θ : A _→ B é ✉♠ Φ✲❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s✱ ❡♥tã♦ θ(A)_{∈ V}✳

❊①❡♠♣❧♦ ✷✳✷✳✷✳ ❆ ✈❛r✐❡❞❛❞❡ q✉❡ ❝♦♥s✐st❡ ❛♣❡♥❛s ♥❛ á❧❣❡❜r❛(0)é ❝❤❛♠❛❞❛ ❞❡ ✈❛r✐❡❞❛❞❡ tr✐✈✐❛❧✳

❊①❡♠♣❧♦ ✷✳✷✳✸✳ ❆ ✈❛r✐❡❞❛❞❡ ❞❡✜♥✐❞❛ ♣❡❧❛ ✐❞❡♥t✐❞❛❞❡ f = (x, y, z) é ❛ ✈❛r✐❡❞❛❞❡ ❞❛s á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s ❡ s❡rá ❞❡♥♦t❛❞❛ ♣♦r Assoc✳

❊①❡♠♣❧♦ ✷✳✷✳✹✳ ❆ ✈❛r✐❡❞❛❞❡ ❞❡✜♥✐❞❛ ♣❡❧❛ ✐❞❡♥t✐❞❛❞❡f = [x, y]é ❛ ✈❛r✐❡❞❛❞❡ ❞❛s á❧❣❡❜r❛s ❝♦♠✉t❛t✐✈❛s ❡ s❡rá ❞❡♥♦t❛❞❛ ♣♦r Com✳

❊①❡♠♣❧♦ ✷✳✷✳✺✳ ❆ ✈❛r✐❡❞❛❞❡ ❞❡✜♥✐❞❛ ♣❡❧❛s ✐❞❡♥t✐❞❛❞❡s f1 = (x, x, y) ❡ f2 = (x, y, y) ❞❡✜♥❡ ❛ ✈❛r✐❡❞❛❞❡ ❞❛s á❧❣❡❜r❛s ❛❧t❡r♥❛t✐✈❛s ❡ s❡rá ❞❡♥♦t❛❞❛ ♣♦rAlt✳

❊①❡♠♣❧♦ ✷✳✷✳✻✳ ❆ ✈❛r✐❡❞❛❞❡ ❞❡✜♥✐❞❛ ♣❡❧❛s ✐❞❡♥t✐❞❛❞❡sf1 = [x, y]❡f2 = (x2_{, y, x)}_{❞❡✜♥❡} ❛ ✈❛r✐❡❞❛❞❡ ❞❛s á❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥ ❡ s❡rá ❞❡♥♦t❛❞❛ ♣♦rJord✳

❊①❡♠♣❧♦ ✷✳✷✳✼✳ ❆ ✈❛r✐❡❞❛❞❡ ❞❡✜♥✐❞❛ ♣❡❧❛s ✐❞❡♥t✐❞❛❞❡s f1 = (x, y, x) ❡ f2 = (x2_{, y, x)} ❞❡✜♥❡ ❛ ✈❛r✐❡❞❛❞❡ ❞❛s á❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥ ♥ã♦ ❝♦♠✉t❛t✐✈❛s ❡ s❡rá ❞❡♥♦t❛❞❛ ♣♦r NCJ✳ ❊①❡♠♣❧♦ ✷✳✷✳✽ ✭✭✲✶✱✶✮✲á❧❣❡❜r❛s✮✳ ❆ ✈❛r✐❡❞❛❞❡ ❞❡✜♥✐❞❛ ♣❡❧❛s ✐❞❡♥t✐❞❛❞❡sf1 = (x, y, z) + (y, z, x) + (z, x, y) ❡ f2 = (x, y, y) ❞❡✜♥❡ ❛ ✈❛r✐❡❞❛❞❡ ❞❛s ✭✲✶✱✶✮✲á❧❣❡❜r❛s ❡ s❡rá ❞❡♥♦t❛❞❛ ♣♦r [₋1_,1_]✶✳

✶_{❆ ✈❛r✐❡❞❛❞❡ ❞❛s ✭✲✶✱✶✮✲á❧❣❡❜r❛s é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ á❧❣❡❜r❛s ❛❧t❡r♥❛t✐✈❛s à ❞✐r❡✐t❛ q✉❡ s❛t✐s❢❛③❡♠ ❛}

(23)

✷✳✸ ❊▲❊▼❊◆❚❖❙ ❉❆ ❚❊❖❘■❆ ●❊❘❆▲ ❉❊ ❘❆❉■❈❆■❙ ✶✶

❊①❡♠♣❧♦ ✷✳✷✳✾ ✭➪❧❣❡❜r❛s ❞❡ ▲✐❡✮✳ ❆ ✈❛r✐❡❞❛❞❡ ❞❡✜♥✐❞❛ ♣❡❧❛s ✐❞❡♥t✐❞❛❞❡s f1 =x(yz) + y(zx) +z(xy)❡ f2 =xy+yx ❞❡✜♥❡ ❛ ✈❛r✐❡❞❛❞❡ ❞❛s á❧❣❡❜r❛s ❞❡ ▲✐❡ ❡ s❡rá ❞❡♥♦t❛❞❛ ♣♦r Lie✳ ❖❜s❡r✈❡ q✉❡ á❧❣❡❜r❛s ❞❡ ▲✐❡ sã♦ ❛♥t✐❝♦♠✉t❛t✐✈❛s✳

❚♦❞♦ ♣♦❧✐♥ô♠✐♦ ♥ã♦ ❛ss♦❝✐❛t✐✈♦ f _∈ Φ[X] ♣♦❞❡ s❡r ✉♥✐❝❛♠❡♥t❡ ❞❡❝♦♠♣♦st♦ ❡♠ ✉♠❛ s♦♠❛ ❞❡ ♠♦♥ô♠✐♦s ✐rr❡❞✉tí✈❡✐s✷_{✳ ❱❛♠♦s ❞✐③❡r q✉❡ ✉♠ ♠♦♥ô♠✐♦} _{αv✱ ❡♠ q✉❡} _α _∈ _Φ _❡ v _∈V[X]✱ t❡♠ t✐♣♦[n1, n2,_{· · ·} , nk]s❡ ❛ ♣❛❧❛✈r❛ ♥ã♦ ❛ss♦❝✐❛t✐✈❛v❝♦♥té♠xi❡①❛t❛♠❡♥t❡ni

✈❡③❡s ❡ t❛♠❜é♠nk 6= 0✱ ♠❛snj = 0✱ ♣❛r❛j > k✳ P♦r ❡①❡♠♣❧♦✱ ♦ ♠♦♥ô♠✐♦((x1x3)x3)(x1x4)

t❡♠ t✐♣♦[2,0,2,1]✳ ❈❤❛♠❛r❡♠♦s ❞❡ni♦ ❣r❛✉ ❞♦ ♠♦♥ô♠✐♦αv❡♠xi✳ ❙❡ t♦❞♦s ♦s ♠♦♥ô♠✐♦s

❡♠ ✉♠❛ ❧✐st❛❣❡♠ ✐rr❡❞✉tí✈❡❧ ❞♦ ♣♦❧✐♥ô♠✐♦f tê♠ ♦ ♠❡s♠♦ ❣r❛✉ni❡♠xi✱ ❡♥tã♦ ❞✐r❡♠♦s q✉❡

♦ ♣♦❧✐♥ô♠✐♦ f é ❤♦♠♦❣ê♥❡♦ ❡♠xi ❞❡ ❣r❛✉ ni✳ ❯♠ ♣♦❧✐♥ô♠✐♦ ♥ã♦ ❛ss♦❝✐❛t✐✈♦ ✭✐❞❡♥t✐❞❛❞❡

❞❡ ✉♠❛ á❧❣❡❜r❛✮ é ❝❤❛♠❛❞♦ ❞❡ ❤♦♠♦❣ê♥❡♦ s❡ t♦❞♦s ♦s s❡✉s ♠♦♥ô♠✐♦s ♥✉♠❛ ❧✐st❛❣❡♠ ✐rr❡❞✉tí✈❡❧ sã♦ ❞♦ ♠❡s♠♦ t✐♣♦✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ✉♠ ♣♦❧✐♥ô♠✐♦ f é ❤♦♠♦❣ê♥❡♦ s❡ ❤á ❤♦♠♦❣❡♥❡✐❞❛❞❡ ❡♠ ❝❛❞❛ ✈❛r✐á✈❡❧✳ P♦r ❡①❡♠♣❧♦✱ ♦ ♣♦❧✐♥ô♠✐♦ (x2

1x3)x4 + ((x1x4)x3)x1 é ❤♦♠♦❣ê♥❡♦✱ ♣♦✐s ❝❛❞❛ ✉♠ ❞♦s s❡✉s ♠♦♥ô♠✐♦s t❡♠ t✐♣♦ [2,0,1,1]✳ ❖ ♣♦❧✐♥ô♠✐♦ x2

1x2 − (x1x3)x1 é ❤♦♠♦❣ê♥❡♦ ❡♠ x1 ❞❡ ❣r❛✉ ✷✱ ♥♦ ❡♥t❛♥t♦✱ ♥ã♦ é ❤♦♠♦❣ê♥❡♦✳ ❯♠ ♣♦❧✐♥ô♠✐♦ ❤♦♠♦❣ê♥❡♦ ❞♦ t✐♣♦ [n1, n2, ..., nk]✱ ❡♠ q✉❡ nj ≤1✱j = 1,· · · , k✱ é ❞✐t♦ s❡r ♠✉❧t✐❧✐♥❡❛r✳

❖ ❣r❛✉ ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦ ♥ã♦ ❛ss♦❝✐❛t✐✈♦ ❡♠xi é ❞❡✜♥✐❞♦ ❝♦♠♦ ♦ ♠á①✐♠♦ ❣r❛✉ ❡♠ xi

❞♦s s❡✉s ♠♦♥ô♠✐♦s✳ ❙❡❥❛ f _∈ Φ[X] ✉♠ ♣♦❧✐♥ô♠✐♦ ❛r❜✐trár✐♦ ♥ã♦ ❛ss♦❝✐❛t✐✈♦✳ ❙❡ r❡❛❣r✉✲ ♣❛r♠♦s ♦s ♠♦♥ô♠✐♦s ❝♦♠ ♦ ♠❡s♠♦ t✐♣♦ ❞♦ ♣♦❧✐♥ô♠✐♦f✱ ❡♥tã♦ f é r❡♣r❡s❡♥t❛❞♦ ♥❛ ❢♦r♠❛ ❞❡ ✉♠❛ s♦♠❛ ❞❡ ♣♦❧✐♥ô♠✐♦s ❤♦♠♦❣ê♥❡♦s✳ ❊st❡s ♣♦❧✐♥ô♠✐♦s ❤♦♠♦❣ê♥❡♦s sã♦ ❝❤❛♠❛❞♦s ❞❡ ❝♦♠♣♦♥❡♥t❡s ❤♦♠♦❣ê♥❡❛s ❞♦ ♣♦❧✐♥ô♠✐♦ f✳

❉❡✜♥✐çã♦ ✷✳✷✳✸ ✭❱❛r✐❡❞❛❞❡ ❤♦♠♦❣ê♥❡❛✮✳ ❙❡❥❛♠ V ✉♠❛ ✈❛r✐❡❞❛❞❡ ❡ f _∈T(_V)✳ ❆ ✈❛r✐❡✲ ❞❛❞❡ V é ❞✐t❛ s❡r ❤♦♠♦❣ê♥❡❛ s❡ t♦❞❛s ❛s ❝♦♠♣♦♥❡♥t❡s ❤♦♠♦❣ê♥❡❛s t❛♠❜é♠ ❡st✐✈❡r❡♠ ❡♠ T(_V)✳

❖s s❡❣✉✐♥t❡s r❡s✉❧t❛❞♦s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬❩❙❙❙✽✷❪✳

▲❡♠❛ ✷✳✷✳✶✵✳ P❛r❛ ❛ ❤♦♠♦❣❡♥❡✐❞❛❞❡ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ V✱ é ♥❡❝❡ssár✐♦ ❡ s✉✜❝✐❡♥t❡ q✉❡ T(_V)✱ ❝♦♠♦ ✉♠ ✐❞❡❛❧ ❞❛ á❧❣❡❜r❛ Φ[X]✱ t❡♥❤❛ ✉♠ s✐st❡♠❛ ❞❡ ❡❧❡♠❡♥t♦s ❣❡r❛❞♦r❡s ❤♦♠♦✲ ❣ê♥❡♦s✳

❚❡♦r❡♠❛ ✷✳✷✳✹✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ á❧❣❡❜r❛s s♦❜r❡ ✉♠ ❝♦r♣♦ ✐♥✜♥✐t♦ é ❤♦♠♦❣ê♥❡❛✳

❯♠❛ ✈❛r✐❡❞❛❞❡ q✉❡ ✐♥t❡r❡ss❛ ❛♦ ❡st✉❞♦✱ ♥❡ss❛ ♣❛rt❡ ✐♥tr♦❞✉tór✐❛✱ é ❛ ✈❛r✐❡❞❛❞❡ ❞❡ á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s✱ ✐st♦ é✱ ❛ ✈❛r✐❡❞❛❞❡ ❞❡✜♥✐❞❛ ♣❡❧❛ ✐❞❡♥t✐❞❛❞❡ f = (x, y, z)✳

◆❛s s❡çõ❡s ✷✳✹ ❡ ✷✳✺✱ t♦r♥❛r❡♠♦s ❛ ❢❛❧❛r ❞❡ ✈❛r✐❡❞❛❞❡s ❡ ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s✳

✷✳✸ ❊❧❡♠❡♥t♦s ❞❛ ❚❡♦r✐❛ ●❡r❛❧ ❞❡ ❘❛❞✐❝❛✐s

◆❛ t❡♦r✐❛ ❞❡ ❛♥é✐s✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ r❛❞✐❝❛❧ é ✉♠ ❞♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❛ s❡r ❞❡s❡♥✈♦❧✈✐❞♦✳ ❙✉♣♦♥❤❛ q✉❡ s❡❥❛ ♥❡❝❡ssár✐♦ ❞❡s❝r❡✈❡r ♦s ❛♥é✐s ❞❡ ❛❧❣✉♠❛ ❝❧❛ss❡ ❞❡ ❛♥é✐s R✱ ♣♦r ❡①❡♠✲ ♣❧♦✱ ❛ ❝❧❛ss❡ ❞❛s Φ✲á❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥✳ ❈♦♠♦ r❡❣r❛✱ ✉♠❛ ❝❧❛ss❡ _C ❞❡ á❧❣❡❜r❛s ❛♣r❡s❡♥t❛